In[1]:= In[2]:= << HolonomicFunctions.m HolonomicFunctions package by Christoph Koutschan, RISC-Linz, Version 1.6 (12.04.2012) --> Type ?HolonomicFunctions for help In[3]:= S[n] Out[3]= S[n] In[4]:= ToOrePolynomials[S[n]] Out[4]= ToOrePolynomials[S[n]] In[5]:= ToOrePolynomial[S[n]] Out[5]= S n In[6]:= ToOrePolynomial[S[n], OreAlgebra[S[n],Der[x]]] Out[6]= S n In[7]:= OreAlgebra[%] Out[7]= K(n, x)[S ; S , 0][D ; 1, D ] n n x x In[8]:= n*%6 Out[8]= (1 + n) S n In[9]:= n**%6 Out[9]= n S n In[10]:= %6**n Out[10]= (1 + n) S n In[11]:= Der[x]**x Out[11]= Der[x] ** x In[12]:= ToOrePolynomial[Der[x]**x] Out[12]= x D + 1 x In[13]:= ToOrePolynomial[x**Der[x]] Out[13]= x D x In[14]:= ToOrePolynomial[(n+3)f[n] - (n+5)f[n+1] + (7n-3)f[n+2], f[n]] 2 Out[14]= (-3 + 7 n) S + (-5 - n) S + (3 + n) n n In[15]:= ToOrePolynomial[(5n+3)f[n] - (n+5)f[n+1] + (2n-5)f[n+2], f[n]] 2 Out[15]= (-5 + 2 n) S + (-5 - n) S + (3 + 5 n) n n In[16]:= {%14} 2 Out[16]= {(-3 + 7 n) S + (-5 - n) S + (3 + n)} n n In[17]:= {%15} 2 Out[17]= {(-5 + 2 n) S + (-5 - n) S + (3 + 5 n)} n n In[18]:= DFinitePlus[{%14}, {%15}] 2 3 4 5 6 4 Out[18]= {(-2904 + 6094 n - 11281 n - 11121 n + 37499 n + 47675 n + 14966 n ) S + n 2 3 4 5 6 3 > (-14280 + 10282 n + 27865 n - 210127 n - 262142 n - 98377 n - 9621 n ) S + n 2 3 4 5 6 2 > (-17592 - 148094 n + 175914 n + 663435 n + 639670 n + 266541 n + 40622 n ) S + n 2 3 4 5 6 > (-83568 - 282464 n - 649165 n - 689178 n - 341575 n - 77700 n - 6414 n ) S + n 2 3 4 5 6 > (40464 + 198432 n + 386107 n + 381342 n + 198724 n + 51922 n + 5345 n )} In[19]:= SetOptions[$Output, PageWidth-> 50]; In[20]:= DFinitePlus[{%14}, {%15}] 2 3 Out[20]= {(-2904 + 6094 n - 11281 n - 11121 n + 4 5 6 4 > 37499 n + 47675 n + 14966 n ) S + n 2 3 > (-14280 + 10282 n + 27865 n - 210127 n - 4 5 6 3 > 262142 n - 98377 n - 9621 n ) S + n 2 > (-17592 - 148094 n + 175914 n + 3 4 5 > 663435 n + 639670 n + 266541 n + 6 2 > 40622 n ) S + n 2 > (-83568 - 282464 n - 649165 n - 3 4 5 6 > 689178 n - 341575 n - 77700 n - 6414 n 2 > ) S + (40464 + 198432 n + 386107 n + n 3 4 5 6 > 381342 n + 198724 n + 51922 n + 5345 n > )} In[21]:= DFiniteTimes[{%14}, {%15}] 2 Out[21]= {(-110880 - 144216 n + 442232 n + 3 4 5 > 674468 n - 40087 n - 268618 n - 6 7 8 4 > 62731 n - 112 n + 588 n ) S + n 2 3 > (38640 - 431672 n - 736885 n + 342469 n + 4 5 6 7 > 192594 n + 21218 n - 1307 n - 535 n - 8 3 > 42 n ) S + (2802240 - 2025276 n - n 2 3 4 > 6946686 n - 2720772 n + 436335 n + 5 6 7 8 > 349254 n + 40620 n - 1866 n - 369 n ) 2 2 > S + (3402560 + 6520472 n + 5204842 n + n 3 4 5 > 2218750 n + 512116 n + 55723 n + 6 7 8 > 881 n - 289 n - 15 n ) S + n 2 > (-713664 - 2611368 n - 3451242 n - 3 4 5 > 2190275 n - 709441 n - 106602 n - 6 7 8 > 2768 n + 965 n + 75 n )} In[22]:= Annihilator[ n!, {S[n]} ] Out[22]= {S + (-1 - n)} n In[23]:= Annihilator[ n! + k!, {S[n], S[k]} ] Out[23]= {k S + n S + (-k - n - k n), n k 2 2 2 > k S + (-1 - 3 k - k ) S + (1 + 2 k + k )} k k In[24]:= UnderTheStaircase[%] Out[24]= {1, S } k In[25]:= ?Ore* OreAction OreAlgebra OreAlgebraGenerators OreAlgebraObject OreAlgebraOperators OreAlgebraPolynomialVariables OreDelta OreGroebnerBasis OreOperatorQ OreOperators OrePlus OrePolynomial OrePolynomialDegree OrePolynomialListCoefficients OrePolynomialSubstitute OrePolynomialToMapleForm OrePolynomialToNonCommutative OrePolynomialToTeXForm OrePolynomialZeroQ OrePower OreReduce OreSigma OreTimes In[26]:= OreGroebnerBasis[%%] Out[26]= {1} In[27]:= OreGroebnerBasis[%23] Out[27]= {k S + n S + (-k - n - k n), n k 2 2 2 > k S + (-1 - 3 k - k ) S + (1 + 2 k + k )} k k In[28]:= Annihilator[ n! + (n+k)! + k!, {S[n], S[k]} ] 2 2 2 2 2 Out[28]= {(k + k n + n ) S + (n - k n - k n ) k 2 3 2 > S + (-k - 3 k - k - n - 3 k n - k n - n 2 2 3 > 3 n - 2 k n - n ) S + k 2 3 2 2 > (k + 2 k + k + 2 k n + 2 k n + n + 2 2 2 3 3 > 3 k n + k n + n + k n ), 2 2 > (k + k n + n ) S S + n k 2 3 2 2 > (-k - 2 k - k - 2 k n - 2 k n - n - 2 2 2 > k n ) S + (-k - n - 2 k n - k n - n 2 2 3 > 2 n - 2 k n - n ) S + k 2 3 2 > (k + 2 k + k + n + 4 k n + 4 k n + 3 2 2 2 2 3 3 > k n + 2 n + 4 k n + 2 k n + n + k n 2 2 2 > ), (k + k n + n ) S + n 2 3 2 > (-k - 3 k - k - n - 3 k n - 2 k n - 2 2 3 2 2 > 3 n - k n - n ) S + (k - k n - k n ) n 2 3 2 3 > S + (k + k + n + 2 k n + 3 k n + k n + k 2 2 2 2 3 > 2 n + 2 k n + k n + n )} In[29]:= UnderTheStaircase[%] Out[29]= {1, S , S } k n In[30]:= OreGroebnerBasis[%28] 2 2 2 2 2 Out[30]= {(k + k n + n ) S + (n - k n - k n ) k 2 3 2 > S + (-k - 3 k - k - n - 3 k n - k n - n 2 2 3 > 3 n - 2 k n - n ) S + k 2 3 2 2 > (k + 2 k + k + 2 k n + 2 k n + n + 2 2 2 3 3 > 3 k n + k n + n + k n ), 2 2 > (k + k n + n ) S S + n k 2 3 2 2 > (-k - 2 k - k - 2 k n - 2 k n - n - 2 2 2 > k n ) S + (-k - n - 2 k n - k n - n 2 2 3 > 2 n - 2 k n - n ) S + k 2 3 2 > (k + 2 k + k + n + 4 k n + 4 k n + 3 2 2 2 2 3 3 > k n + 2 n + 4 k n + 2 k n + n + k n 2 2 2 > ), (k + k n + n ) S + n 2 3 2 > (-k - 3 k - k - n - 3 k n - 2 k n - 2 2 3 2 2 > 3 n - k n - n ) S + (k - k n - k n ) n 2 3 2 3 > S + (k + k + n + 2 k n + 3 k n + k n + k 2 2 2 2 3 > 2 n + 2 k n + k n + n )} In[31]:= UnderTheStaircase[%] Out[31]= {1, S , S } k n In[32]:= OreGroebnerBasis[%28, MonomialOrder -> Lexicographic] 2 3 Out[32]= {(1 - k - k n) S + k 2 3 2 2 > (-5 + 6 k + 2 k + 6 k n + 3 k n + k n ) 2 2 3 4 > S + (4 - 2 k - 10 k - 6 k - k - n - k 2 3 2 2 > 10 k n - 9 k n - 2 k n - n - 3 k n - 2 2 2 3 4 > k n ) S + (2 k + 5 k + 4 k + k + n + k 2 3 2 2 > 5 k n + 6 k n + 2 k n + n + 2 k n + 2 2 2 2 > k n ), (n - k n - k n ) S + n 2 2 2 > (k + k n + n ) S + k 2 3 2 2 > (-k - 3 k - k - n - 3 k n - k n - 3 n - 2 3 > 2 k n - n ) S + k 2 3 2 2 > (k + 2 k + k + 2 k n + 2 k n + n + 2 2 2 3 3 > 3 k n + k n + n + k n )} In[33]:= UnderTheStaircase[%] 2 Out[33]= {1, S , S } k k In[34]:= ann1 = Annihilator[ n! + (n+k)! + k!, {S[n], S[k]} ]; In[35]:= ann2 = Annihilator[ (2n)! + (n-k)!, {S[n], S[k]} ]; In[36]:= Printlevel Out[36]= 0 In[37]:= Printlevel = 5 Out[37]= 5 In[38]:= DFiniteTimes[ann1, ann2] Monomials: {1} Monomials: {S[k], S[n]} Monomials: {S[n], S[k]^2, S[k]*S[n]} Monomials: {S[k]^2, S[k]*S[n], S[n]^2} Monomials: {S[k]*S[n], S[n]^2, S[k]^3,\ > S[k]^2*S[n]} Monomials: {S[n]^2, S[k]^3, S[k]^2*S[n],\ > S[k]*S[n]^2} Monomials: {S[k]^3, S[k]^2*S[n],\ > S[k]*S[n]^2, S[n]^3} Monomials: {S[k]^2*S[n], S[k]*S[n]^2,\ > S[n]^3} Monomials: {S[k]*S[n]^2, S[n]^3} Monomials: {S[n]^3} 3 4 5 6 Out[38]= {(-64 k - 632 k - 1976 k - 3318 k - 7 8 9 10 > 4052 k - 4174 k - 3280 k - 1644 k - 11 12 13 14 > 432 k - 26 k + 12 k + 2 k - 2 3 > 176 k n - 896 k n - 2472 k n - 4 5 6 > 7980 k n - 19983 k n - 30207 k n - 7 8 9 > 29762 k n - 22769 k n - 14822 k n - 10 11 12 > 7095 k n - 1892 k n - 111 k n + 13 14 2 > 59 k n + 10 k n + 176 n - 2 2 2 3 2 > 1136 k n - 6656 k n - 14444 k n - 4 2 5 2 6 2 > 29937 k n - 65490 k n - 98067 k n - 7 2 8 2 9 2 > 88540 k n - 50947 k n - 23003 k n - 10 2 11 2 12 2 > 9685 k n - 2840 k n - 228 k n + 13 2 14 2 3 > 89 k n + 16 k n + 2096 n - 3 2 3 3 3 > 504 k n - 16372 k n - 27154 k n - 4 3 5 3 > 29261 k n - 67068 k n - 6 3 7 3 > 131591 k n - 135278 k n - 8 3 9 3 10 3 > 68452 k n - 15150 k n - 2231 k n - 11 3 12 3 13 3 > 1290 k n - 281 k n + 38 k n + 14 3 4 4 > 8 k n + 10264 n + 14948 k n - 2 4 3 4 4 4 > 7348 k n + 3588 k n + 71221 k n + 5 4 6 4 7 4 > 84909 k n - 12308 k n - 98870 k n - 8 4 9 4 10 4 > 70161 k n - 9071 k n + 6004 k n + 11 4 12 4 13 4 > 1246 k n - 102 k n + 12 k n + 5 5 2 5 > 25824 n + 55969 k n + 33987 k n + 3 5 4 5 > 72106 k n + 213084 k n + 5 5 6 5 > 285092 k n + 182615 k n + 7 5 8 5 9 5 > 7840 k n - 55233 k n - 16845 k n + 10 5 11 5 12 5 > 4915 k n + 1788 k n - 28 k n + 13 5 6 6 > 16 k n + 31771 n + 86434 k n + 2 6 3 6 > 49843 k n + 71426 k n + 4 6 5 6 > 208028 k n + 281756 k n + 6 6 7 6 > 228208 k n + 83656 k n - 8 6 9 6 10 6 > 23516 k n - 21173 k n + 310 k n + 11 6 12 6 7 > 1096 k n - 64 k n + 5001 n + 7 2 7 3 7 > 42916 k n - 34591 k n - 43314 k n + 4 7 5 7 > 67991 k n + 110254 k n + 6 7 7 7 8 7 > 108022 k n + 76098 k n + 6216 k n - 9 7 10 7 11 7 > 13882 k n - 1692 k n + 528 k n - 8 8 2 8 > 37015 n - 43923 k n - 166048 k n - 3 8 4 8 > 177330 k n - 28080 k n + 5 8 6 8 7 8 > 14410 k n + 5516 k n + 30544 k n + 8 8 9 8 10 8 > 16264 k n - 3996 k n - 1344 k n + 11 8 9 9 > 64 k n - 44725 n - 55006 k n - 2 9 3 9 > 188648 k n - 230496 k n - 4 9 5 9 6 9 > 59657 k n + 18151 k n - 8308 k n - 7 9 8 9 9 9 > 16 k n + 9496 k n + 496 k n - 10 9 10 10 > 256 k n - 13753 n + 20567 k n - 2 10 3 10 > 65335 k n - 171264 k n - 4 10 5 10 > 76898 k n + 10523 k n + 6 10 7 10 8 10 > 4132 k n - 5488 k n + 1728 k n + 9 10 11 11 > 320 k n + 10901 n + 64804 k n + 2 11 3 11 > 58721 k n - 54960 k n - 4 11 5 11 > 55709 k n - 9542 k n + 6 11 7 11 8 11 > 4528 k n - 1696 k n - 64 k n + 12 12 2 12 > 9719 n + 34281 k n + 72886 k n + 3 12 4 12 > 14786 k n - 12430 k n - 5 12 6 12 7 12 > 9420 k n + 832 k n - 64 k n + 13 13 2 13 > 1305 n - 3805 k n + 27463 k n + 3 13 4 13 > 15444 k n + 5428 k n - 5 13 6 13 14 > 2256 k n - 64 k n - 1174 n - 14 2 14 3 14 > 9903 k n + 1014 k n + 1448 k n + 4 14 5 14 15 > 2944 k n - 64 k n - 418 n - 15 2 15 3 15 > 3046 k n - 1372 k n - 1264 k n + 4 15 16 16 > 320 k n + 12 n - 36 k n - 3 16 17 17 2 17 > 256 k n + 16 n + 80 k n + 64 k n 3 3 4 5 > ) S + (30 k + 242 k + 826 k + k 6 7 8 9 > 1604 k + 1968 k + 1580 k + 826 k + 10 11 12 > 270 k + 50 k + 4 k - 30 k n - 2 3 4 > 197 k n - 546 k n - 715 k n - 5 6 7 > 14 k n + 1678 k n + 3228 k n + 8 9 10 > 3291 k n + 2036 k n + 761 k n + 11 12 2 2 > 158 k n + 14 k n + 30 n - 122 k n - 2 2 3 2 4 2 > 1012 k n - 2206 k n - 2507 k n - 5 2 6 2 7 2 > 2144 k n - 1918 k n - 1440 k n - 8 2 9 2 10 2 > 464 k n + 221 k n + 255 k n + 11 2 12 2 3 > 85 k n + 10 k n + 301 n + 3 2 3 3 3 > 482 k n - 1183 k n - 2897 k n - 4 3 5 3 6 3 > 902 k n + 2835 k n + 3713 k n + 7 3 8 3 9 3 > 1627 k n - 182 k n - 426 k n - 10 3 11 3 4 > 141 k n - 15 k n + 986 n + 4 2 4 3 4 > 2820 k n + 1914 k n - 614 k n - 4 4 5 4 6 4 > 193 k n + 2383 k n + 3829 k n + 7 4 8 4 9 4 > 2987 k n + 1340 k n + 373 k n + 10 4 11 4 5 > 71 k n + 8 k n + 1250 n + 5 2 5 3 5 > 3907 k n + 4007 k n + 2774 k n + 4 5 5 5 6 5 > 2390 k n + 1549 k n + 440 k n - 7 5 8 5 9 5 > 285 k n - 383 k n - 162 k n - 10 5 6 6 > 24 k n + 233 n + 752 k n - 2 6 3 6 4 6 > 1598 k n - 2811 k n - 1051 k n + 5 6 6 6 7 6 > 839 k n + 1497 k n + 943 k n + 8 6 9 6 7 > 274 k n + 28 k n - 957 n - 7 2 7 3 7 > 1861 k n - 4490 k n - 5525 k n - 4 7 5 7 6 7 > 4540 k n - 2703 k n - 975 k n - 7 7 8 7 8 > 148 k n + 4 k n - 903 n - 8 2 8 3 8 > 1370 k n - 1943 k n - 1235 k n - 4 8 5 8 6 8 > 525 k n - 255 k n - 152 k n - 7 8 9 9 2 9 > 36 k n - 12 n + 293 k n + 113 k n - 3 9 4 9 5 9 > 149 k n - 123 k n + 2 k n + 6 9 10 10 > 4 k n + 243 n + 690 k n + 2 10 3 10 4 10 > 936 k n + 508 k n + 156 k n + 5 10 11 11 > 20 k n - 16 n - 83 k n + 2 11 3 11 4 11 > 4 k n - 5 k n + 12 k n - 12 12 2 12 > 19 n - 54 k n - 39 k n - 3 12 13 13 2 13 2 > 20 k n + 4 n + 8 k n + 4 k n ) S n 2 3 4 5 > + (300 k + 2450 k + 7602 k + 11016 k + 6 7 8 9 > 5928 k - 4770 k - 10446 k - 7952 k - 10 11 12 13 > 3276 k - 758 k - 90 k - 4 k + 2 3 > 360 k n + 3176 k n + 15129 k n + 4 5 6 > 39762 k n + 58101 k n + 44320 k n + 7 8 9 > 3577 k n - 25917 k n - 24897 k n - 10 11 12 > 11222 k n - 2723 k n - 342 k n - 13 14 2 2 > 21 k n - k n - 660 n + 962 k n + 2 2 3 2 4 2 > 12955 k n + 42480 k n + 78876 k n + 5 2 6 2 7 2 > 96926 k n + 86670 k n + 45407 k n - 8 2 9 2 10 2 > 1299 k n - 18266 k n - 11106 k n - 11 2 12 2 13 2 > 2979 k n - 366 k n - 22 k n - 14 2 3 3 > 2 k n - 6444 n - 8825 k n + 2 3 3 3 4 3 > 13302 k n + 54751 k n + 67208 k n + 5 3 6 3 7 3 > 35917 k n + 21154 k n + 28506 k n + 8 3 9 3 10 3 > 18099 k n + 534 k n - 3720 k n - 11 3 12 3 13 3 > 1361 k n - 138 k n + 3 k n - 4 4 2 4 > 25589 n - 50556 k n - 36013 k n - 3 4 4 4 5 4 > 12869 k n - 29874 k n - 72839 k n - 6 4 7 4 8 4 > 76633 k n - 31895 k n - 325 k n + 9 4 10 4 11 4 > 1724 k n - 839 k n - 495 k n - 12 4 13 4 5 > 68 k n - 4 k n - 51548 n - 5 2 5 3 5 > 106216 k n - 89159 k n - 97306 k n - 4 5 5 5 > 141269 k n - 136258 k n - 6 5 7 5 8 5 > 91453 k n - 34837 k n - 2783 k n + 9 5 10 5 11 5 > 941 k n - 383 k n - 113 k n + 12 5 6 6 > 16 k n - 48203 n - 102899 k n - 2 6 3 6 4 6 > 31657 k n - 23694 k n - 95968 k n - 5 6 6 6 7 6 > 99655 k n - 64799 k n - 22390 k n + 8 6 9 6 10 6 > 2102 k n + 2674 k n - 70 k n - 11 6 7 7 > 140 k n + 3923 n - 20460 k n + 2 7 3 7 > 91581 k n + 115898 k n + 4 7 5 7 6 7 > 37092 k n + 1607 k n - 17880 k n - 7 7 8 7 9 7 > 16568 k n - 1972 k n + 1718 k n + 10 7 11 7 8 > 316 k n - 16 k n + 53451 n + 8 2 8 > 49418 k n + 141465 k n + 3 8 4 8 > 130397 k n + 51895 k n + 5 8 6 8 7 8 > 29340 k n + 12022 k n - 4264 k n - 8 8 9 8 10 8 > 2900 k n - 16 k n + 64 k n + 9 9 2 9 > 44191 n + 29715 k n + 94299 k n + 3 9 4 9 5 9 > 78884 k n + 13632 k n + 1019 k n + 6 9 7 9 8 9 > 6973 k n + 1510 k n - 616 k n - 9 9 10 10 > 80 k n + 5083 n - 25111 k n + 2 10 3 10 > 15657 k n + 36457 k n + 4 10 5 10 6 10 > 10254 k n - 4628 k n - 290 k n + 7 10 8 10 11 > 656 k n + 16 k n - 10231 n - 11 2 11 > 32801 k n - 22760 k n + 3 11 4 11 5 11 > 4289 k n + 7690 k n + 623 k n - 6 11 7 11 12 > 416 k n + 16 k n - 4200 n - 12 2 12 3 12 > 8852 k n - 14255 k n - 5385 k n + 4 12 5 12 6 12 > 488 k n + 628 k n + 16 k n + 13 13 2 13 > 301 n + 2485 k n - 1389 k n - 3 13 4 13 5 13 > 1605 k n - 648 k n + 16 k n + 14 14 2 14 > 298 n + 1368 k n + 682 k n + 3 14 4 14 15 > 188 k n - 80 k n - 28 n + 15 2 15 3 15 > 40 k n + 68 k n + 64 k n - 16 16 2 16 > 16 n - 32 k n - 16 k n ) S S + n k 3 4 5 6 > (140 k + 1070 k + 4220 k + 9750 k + 7 8 9 > 15080 k + 17096 k + 14318 k + 10 11 12 13 > 8366 k + 3096 k + 598 k + 12 k - 14 15 2 > 16 k - 2 k - 440 k n - 2980 k n - 3 4 5 > 8902 k n - 11258 k n + 10803 k n + 6 7 8 > 59969 k n + 99563 k n + 100936 k n + 9 10 11 > 73081 k n + 38719 k n + 13757 k n + 12 13 14 > 2604 k n + 20 k n - 82 k n - 15 2 2 > 10 k n + 440 n - 2260 k n - 2 2 3 2 > 17990 k n - 57648 k n - 4 2 5 2 > 104191 k n - 68564 k n + 6 2 7 2 > 88169 k n + 228218 k n + 8 2 9 2 > 229336 k n + 141781 k n + 10 2 11 2 > 63485 k n + 21208 k n + 12 2 13 2 14 2 > 4219 k n + 79 k n - 132 k n - 15 2 3 3 > 16 k n + 5100 n - 1058 k n - 2 3 3 3 > 43440 k n - 139402 k n - 4 3 5 3 > 294743 k n - 344735 k n - 6 3 7 3 > 115478 k n + 186889 k n + 8 3 9 3 > 263363 k n + 144792 k n + 10 3 11 3 > 43432 k n + 11043 k n + 12 3 13 3 14 3 > 2948 k n + 283 k n - 62 k n - 15 3 4 4 > 8 k n + 26880 n + 25628 k n - 2 4 3 4 > 49984 k n - 148230 k n - 4 4 5 4 > 369057 k n - 610001 k n - 6 4 7 4 > 492569 k n - 90524 k n + 8 4 9 4 > 170110 k n + 116291 k n + 10 4 11 4 12 4 > 13449 k n - 5506 k n - 525 k n + 13 4 14 4 5 > 140 k n - 12 k n + 82498 n + 5 2 5 3 5 > 120031 k n - 3711 k n + 7617 k n - 4 5 5 5 > 95821 k n - 484905 k n - 6 5 7 5 > 642434 k n - 369696 k n + 8 5 9 5 > 29875 k n + 106947 k n + 10 5 11 5 > 18342 k n - 7688 k n - 12 5 13 5 14 5 > 1700 k n - 8 k n - 16 k n + 6 6 2 6 > 152993 n + 291306 k n + 87519 k n + 3 6 4 6 > 248928 k n + 387663 k n + 5 6 6 6 > 7260 k n - 368135 k n - 7 6 8 6 > 419328 k n - 101466 k n + 9 6 10 6 > 72883 k n + 28291 k n - 11 6 12 6 13 6 > 2090 k n - 904 k n + 64 k n + 7 7 2 7 > 148893 n + 415531 k n + 127800 k n + 3 7 4 7 > 371561 k n + 642964 k n + 5 7 6 7 > 363016 k n + 44996 k n - 7 7 8 7 > 221512 k n - 146266 k n + 9 7 10 7 11 7 > 16155 k n + 20390 k n + 572 k n - 12 7 8 8 > 528 k n - 1703 n + 295921 k n - 2 8 3 8 > 29483 k n + 301802 k n + 4 8 5 8 > 534471 k n + 258450 k n + 6 8 7 8 8 8 > 164338 k n - 4096 k n - 91820 k n - 9 8 10 8 11 8 > 14806 k n + 7692 k n + 1232 k n - 12 8 9 9 > 64 k n - 194821 n - 20946 k n - 2 9 3 9 > 388991 k n + 126179 k n + 4 9 5 9 > 405922 k n + 57820 k n + 6 9 7 9 8 9 > 26436 k n + 55118 k n - 23324 k n - 9 9 10 9 11 9 > 13276 k n + 80 k n + 256 k n - 10 10 > 228993 n - 185527 k n - 2 10 3 10 > 674682 k n - 119912 k n + 4 10 5 10 > 337743 k n + 81187 k n - 6 10 7 10 > 44582 k n + 22228 k n + 8 10 9 10 10 10 > 3472 k n - 2704 k n - 320 k n - 11 11 2 11 > 91509 n - 37712 k n - 588356 k n - 3 11 4 11 > 309635 k n + 142698 k n + 5 11 6 11 > 117809 k n - 12474 k n + 7 11 8 11 9 11 > 288 k n + 2208 k n + 64 k n + 12 12 > 33817 n + 155759 k n - 2 12 3 12 > 239611 k n - 255828 k n - 4 12 5 12 > 40381 k n + 51656 k n + 6 12 7 12 8 12 > 6620 k n - 864 k n + 64 k n + 13 13 2 13 > 49047 n + 159669 k n + 4912 k n - 3 13 4 13 > 76268 k n - 56628 k n + 5 13 6 13 7 13 > 1680 k n + 2640 k n + 64 k n + 14 14 2 14 > 16992 n + 61619 k n + 38775 k n + 3 14 4 14 > 10694 k n - 14200 k n - 5 14 6 14 15 > 3040 k n + 64 k n + 676 n + 15 2 15 3 15 > 5449 k n + 8146 k n + 10028 k n + 4 15 5 15 16 > 240 k n - 320 k n - 474 n - 16 2 16 3 16 > 2270 k n - 1692 k n + 1168 k n + 4 16 17 17 > 256 k n + 116 n - 300 k n - 2 17 3 17 18 18 > 528 k n - 64 k n + 48 n + 48 k n 2 2 3 > ) S + (-300 k - 3410 k - 16786 k - k 4 5 6 > 47654 k - 87450 k - 109374 k - 7 8 9 > 94998 k - 56794 k - 22310 k - 10 11 12 13 > 4970 k - 206 k + 198 k + 50 k + 14 2 > 4 k + 300 n + 1908 k n - 2000 k n - 3 4 5 > 41746 k n - 149015 k n - 295710 k n - 6 7 8 > 386232 k n - 348530 k n - 217657 k n - 9 10 11 > 90974 k n - 23055 k n - 2332 k n + 12 13 14 > 410 k n + 148 k n + 13 k n + 2 2 2 2 > 1358 n + 18190 k n + 55248 k n + 3 2 4 2 > 48882 k n - 67543 k n - 5 2 6 2 > 250659 k n - 382669 k n - 7 2 8 2 > 377744 k n - 256353 k n - 9 2 10 2 > 118351 k n - 35328 k n - 11 2 12 2 13 2 > 5947 k n - 285 k n + 65 k n + 14 2 3 3 > 8 k n - 1828 n + 40162 k n + 2 3 3 3 > 175294 k n + 269628 k n + 4 3 5 3 > 183798 k n + 13041 k n - 6 3 7 3 > 106012 k n - 140052 k n - 8 3 9 3 > 110075 k n - 57641 k n - 10 3 11 3 12 3 > 19633 k n - 4048 k n - 434 k n - 13 3 4 4 > 16 k n - 23810 n - 244 k n + 2 4 3 4 > 205562 k n + 385416 k n + 4 4 5 4 > 279615 k n + 49466 k n - 6 4 7 4 8 4 > 67221 k n - 65696 k n - 36587 k n - 9 4 10 4 11 4 > 16099 k n - 4933 k n - 778 k n - 12 4 13 4 5 > 25 k n + 4 k n - 63075 n - 5 2 5 > 131902 k n + 25358 k n + 3 5 4 5 > 226702 k n + 159131 k n - 5 5 6 5 > 51634 k n - 139319 k n - 7 5 8 5 9 5 > 88653 k n - 31859 k n - 9899 k n - 10 5 11 5 12 5 > 2803 k n - 432 k n - 16 k n - 6 6 2 6 > 78053 n - 210583 k n - 137715 k n + 3 6 4 6 5 6 > 28408 k n + 41083 k n - 71790 k n - 6 6 7 6 > 118025 k n - 60469 k n - 8 6 9 6 10 6 > 12096 k n - 1396 k n - 585 k n - 11 6 7 7 > 116 k n - 39948 n - 116549 k n - 2 7 3 7 4 7 > 54318 k n + 56660 k n + 90517 k n + 5 7 6 7 7 7 > 10150 k n - 53600 k n - 33257 k n - 8 7 9 7 10 7 > 4854 k n + 692 k n + 84 k n - 11 7 8 8 > 16 k n + 14827 n + 35632 k n + 2 8 3 8 > 99647 k n + 139476 k n + 4 8 5 8 6 8 > 153022 k n + 82623 k n + 4809 k n - 7 8 8 8 9 8 > 11440 k n - 2962 k n + 32 k n + 10 8 9 9 > 32 k n + 34836 n + 87212 k n + 2 9 3 9 > 112742 k n + 96891 k n + 4 9 5 9 6 9 > 96488 k n + 63173 k n + 18285 k n - 7 9 8 9 9 9 > 44 k n - 664 k n - 32 k n + 10 10 2 10 > 20330 n + 44839 k n + 35143 k n + 3 10 4 10 > 14628 k n + 17510 k n + 5 10 6 10 7 10 > 17169 k n + 7270 k n + 816 k n - 8 10 11 11 > 32 k n + 4150 n + 3729 k n - 2 11 3 11 > 9167 k n - 12667 k n - 4 11 5 11 6 11 > 7812 k n - 572 k n + 880 k n + 7 11 12 12 > 128 k n - 375 n - 4098 k n - 2 12 3 12 4 12 > 7820 k n - 4806 k n - 3749 k n - 5 12 6 12 13 > 1028 k n - 32 k n - 111 n - 13 2 13 3 13 > 1210 k n - 1127 k n + 380 k n - 4 13 5 13 14 > 280 k n - 96 k n + 59 n - 14 2 14 3 14 > 94 k n + 43 k n + 292 k n + 4 14 15 15 2 15 > 32 k n + 12 n - 8 k n - 4 k n + 3 15 > 16 k n ) S + n 2 3 4 5 > (300 k + 3282 k + 14266 k + 35166 k + 6 7 8 > 56072 k + 62700 k + 49974 k + 9 10 11 > 27364 k + 9338 k + 1386 k - 12 13 14 15 > 260 k - 164 k - 30 k - 2 k - 2 3 > 952 k n - 3214 k n + 12990 k n + 4 5 6 > 99028 k n + 267050 k n + 410916 k n + 7 8 9 > 420588 k n + 302818 k n + 151713 k n + 10 11 12 > 48775 k n + 7399 k n - 900 k n - 13 14 15 > 649 k n - 115 k n - 7 k n + 2 2 2 2 > 652 n - 9514 k n - 52398 k n - 3 2 4 2 > 70446 k n + 157708 k n + 5 2 6 2 > 722323 k n + 1202475 k n + 7 2 8 2 > 1184660 k n + 772062 k n + 9 2 10 2 > 339500 k n + 95426 k n + 11 2 12 2 13 2 > 12781 k n - 1482 k n - 941 k n - 14 2 15 2 3 > 139 k n - 5 k n + 9446 n - 3 2 3 > 36062 k n - 248094 k n - 3 3 4 3 > 557246 k n - 412327 k n + 5 3 6 3 > 600009 k n + 1656728 k n + 7 3 8 3 > 1779758 k n + 1104059 k n + 9 3 10 3 > 414806 k n + 88710 k n + 11 3 12 3 13 3 > 6264 k n - 2426 k n - 907 k n - 14 3 4 4 > 108 k n + 61204 n - 47032 k n - 2 4 3 4 > 585978 k n - 1498584 k n - 4 4 5 4 > 1921984 k n - 900066 k n + 6 4 7 4 > 709531 k n + 1414151 k n + 8 4 9 4 > 973329 k n + 320451 k n + 10 4 11 4 > 36661 k n - 5637 k n - 12 4 13 4 14 4 > 2309 k n - 487 k n - 66 k n + 5 5 2 5 > 231378 n + 106388 k n - 733752 k n - 3 5 4 5 > 2088852 k n - 3069960 k n - 5 5 6 5 > 2510717 k n - 897372 k n + 7 5 8 5 > 385438 k n + 559705 k n + 9 5 10 5 > 198588 k n + 3373 k n - 11 5 12 5 13 5 > 11277 k n - 1597 k n - 20 k n - 14 5 6 6 > 12 k n + 556006 n + 587529 k n - 2 6 3 6 > 281075 k n - 1537706 k n - 4 6 5 6 > 2500132 k n - 2360160 k n - 6 6 7 6 > 1394309 k n - 287289 k n + 8 6 9 6 > 218060 k n + 137937 k n + 10 6 11 6 > 10824 k n - 7632 k n - 12 6 13 6 7 > 1320 k n + 16 k n + 861195 n + 7 2 7 > 1211287 k n + 667818 k n - 3 7 4 7 > 274460 k n - 1089223 k n - 5 7 6 7 > 1093156 k n - 751171 k n - 7 7 8 7 > 334698 k n + 19967 k n + 9 7 10 7 11 7 > 78865 k n + 19865 k n - 896 k n - 12 7 8 8 > 528 k n + 807578 n + 1351660 k n + 2 8 3 8 > 1420553 k n + 777657 k n - 4 8 5 8 > 276097 k n - 399013 k n - 6 8 7 8 > 159176 k n - 141784 k n - 8 8 9 8 10 8 > 54954 k n + 13996 k n + 9758 k n + 11 8 12 8 9 > 848 k n - 80 k n + 316684 n + 9 2 9 > 656714 k n + 1361711 k n + 3 9 4 9 > 1254265 k n + 26201 k n - 5 9 6 9 7 9 > 343009 k n - 39935 k n - 4697 k n - 8 9 9 9 10 9 > 30702 k n - 7380 k n + 1044 k n + 11 9 10 10 > 256 k n - 214360 n - 315662 k n + 2 10 3 10 > 623509 k n + 1130508 k n + 4 10 5 10 > 278336 k n - 233563 k n - 6 10 7 10 > 74623 k n + 15807 k n - 8 10 9 10 10 10 > 656 k n - 2656 k n - 240 k n - 11 11 > 384822 n - 743904 k n - 2 11 3 11 > 73951 k n + 594518 k n + 4 11 5 11 > 335090 k n - 33625 k n - 6 11 7 11 8 11 > 49502 k n + 112 k n + 2128 k n - 12 12 > 246446 n - 545114 k n - 2 12 3 12 > 273147 k n + 116223 k n + 4 12 5 12 > 168467 k n + 36765 k n - 6 12 7 12 8 12 > 10038 k n - 2000 k n + 160 k n - 13 13 > 80754 n - 208374 k n - 2 13 3 13 > 149331 k n - 44691 k n + 4 13 5 13 > 30099 k n + 16980 k n + 6 13 7 13 14 > 492 k n - 256 k n - 11862 n - 14 2 14 > 38121 k n - 29502 k n - 3 14 4 14 > 29770 k n - 2984 k n + 5 14 6 14 15 > 2080 k n + 224 k n - 735 n - 15 2 15 3 15 > 813 k n + 3025 k n - 5680 k n - 4 15 16 16 > 1408 k n - 612 n + 510 k n + 2 16 3 16 4 16 > 2266 k n - 320 k n - 80 k n - 17 17 2 17 > 264 n - 44 k n + 332 k n - 18 18 2 18 > 32 n - 16 k n + 16 k n ) S + k 2 3 4 > (600 k + 5920 k + 23874 k + 53784 k + 5 6 7 > 75810 k + 67396 k + 31344 k - 8 9 10 > 4256 k - 17526 k - 12528 k - 11 12 13 14 > 4522 k - 720 k + 48 k + 36 k + 15 2 > 4 k - 600 n + 872 k n + 39074 k n + 3 4 5 > 184354 k n + 427628 k n + 604820 k n + 6 7 8 > 553882 k n + 307421 k n + 56355 k n - 9 10 11 > 58534 k n - 55845 k n - 22626 k n - 12 13 14 > 4353 k n - 52 k n + 131 k n + 15 2 2 > 17 k n - 6504 n - 26522 k n + 2 2 3 2 > 64490 k n + 552390 k n + 4 2 5 2 > 1396946 k n + 1992225 k n + 6 2 7 2 > 1836009 k n + 1100072 k n + 8 2 9 2 > 358645 k n - 18760 k n - 10 2 11 2 > 81994 k n - 40473 k n - 12 2 13 2 14 2 > 9535 k n - 789 k n + 107 k n + 15 2 3 3 > 21 k n - 31290 n - 171578 k n - 2 3 3 3 > 163194 k n + 716134 k n + 4 3 5 3 > 2388893 k n + 3519432 k n + 6 3 7 3 > 3200193 k n + 1924653 k n + 8 3 9 3 > 723678 k n + 111372 k n - 10 3 11 3 > 41815 k n - 31217 k n - 12 3 13 3 14 3 > 8897 k n - 1186 k n - 22 k n + 15 3 4 4 > 8 k n - 88874 n - 515608 k n - 2 4 3 4 > 916178 k n + 4748 k n + 4 4 5 4 > 2191304 k n + 3646686 k n + 6 4 7 4 > 3252412 k n + 1843959 k n + 8 4 9 4 > 663770 k n + 127281 k n - 10 4 11 4 > 4094 k n - 10489 k n - 12 4 13 4 14 4 > 3272 k n - 499 k n - 30 k n - 5 5 2 5 > 165730 n - 940114 k n - 1913174 k n - 3 5 4 5 > 1334441 k n + 814601 k n + 5 5 6 5 > 2321742 k n + 2104475 k n + 7 5 8 5 > 1064802 k n + 301821 k n + 9 5 10 5 11 5 > 30548 k n - 9304 k n - 4167 k n - 12 5 13 5 14 5 > 687 k n - 30 k n + 4 k n - 6 6 > 207964 n - 1126445 k n - 2 6 3 6 > 2379665 k n - 2150734 k n - 4 6 5 6 > 379287 k n + 998240 k n + 6 6 7 6 > 1044708 k n + 469049 k n + 8 6 9 6 10 6 > 91134 k n - 6129 k n - 8759 k n - 11 6 12 6 13 6 > 2812 k n - 470 k n - 28 k n - 7 7 2 7 > 159937 n - 878612 k n - 1972251 k n - 3 7 4 7 > 1964279 k n - 720022 k n + 5 7 6 7 > 392480 k n + 545877 k n + 7 7 8 7 9 7 > 221834 k n + 34887 k n + 2138 k n + 10 7 11 7 12 7 > 317 k n - 218 k n - 76 k n - 8 8 2 8 > 28488 n - 350994 k n - 1122402 k n - 3 8 4 8 > 1354379 k n - 678830 k n + 5 8 6 8 > 151197 k n + 325812 k n + 7 8 8 8 9 8 > 118246 k n + 7846 k n - 890 k n + 10 8 11 8 12 8 > 832 k n + 116 k n - 16 k n + 9 9 2 9 > 103614 n + 121906 k n - 377468 k n - 3 9 4 9 > 831210 k n - 628592 k n - 5 9 6 9 > 46056 k n + 164031 k n + 7 9 8 9 9 9 > 68977 k n + 2674 k n - 2180 k n + 10 9 11 9 10 > 40 k n + 48 k n + 151384 n + 10 2 10 > 328390 k n + 64372 k n - 3 10 4 10 > 404696 k n - 474827 k n - 5 10 6 10 > 154935 k n + 36031 k n + 7 10 8 10 9 10 > 31677 k n + 3628 k n - 632 k n - 10 10 11 11 > 64 k n + 111326 n + 286558 k n + 2 11 3 11 > 214698 k n - 75505 k n - 4 11 5 11 > 209763 k n - 116632 k n - 6 11 7 11 > 15236 k n + 6700 k n + 8 11 12 12 > 1512 k n + 47926 n + 148346 k n + 2 12 3 12 > 162678 k n + 56079 k n - 4 12 5 12 > 25890 k n - 36607 k n - 6 12 7 12 8 12 > 11174 k n - 120 k n + 160 k n + 13 13 2 13 > 11020 n + 45024 k n + 60166 k n + 3 13 4 13 > 39821 k n + 16397 k n - 5 13 6 13 7 13 > 2254 k n - 2092 k n - 160 k n + 14 14 2 14 > 1020 n + 6461 k n + 7589 k n + 3 14 4 14 5 14 > 6906 k n + 6570 k n + 1044 k n - 6 14 15 15 > 64 k n + 237 n + 280 k n - 2 15 3 15 4 15 > 1427 k n - 1050 k n + 612 k n + 5 15 16 16 > 128 k n + 172 n + 156 k n - 2 16 3 16 4 16 > 380 k n - 380 k n - 16 k n + 17 17 3 17 > 32 n + 48 k n - 16 k n ), 3 4 5 6 > (-32 k - 268 k - 570 k - 670 k - 7 8 9 10 > 736 k - 648 k - 300 k - 48 k + 11 12 2 > 6 k + 2 k - 88 k n - 316 k n - 3 4 5 > 734 k n - 2849 k n - 5486 k n - 6 7 8 > 5366 k n - 3837 k n - 2567 k n - 9 10 11 > 1100 k n - 164 k n + 27 k n + 12 2 2 2 2 > 8 k n + 88 n - 744 k n - 2370 k n - 3 2 4 2 5 2 > 3492 k n - 9745 k n - 18171 k n - 6 2 7 2 8 2 > 16660 k n - 8093 k n - 3070 k n - 9 2 10 2 11 2 > 1146 k n - 185 k n + 34 k n + 12 2 3 3 > 8 k n + 1092 n - 2394 k n - 2 3 3 3 4 3 > 6402 k n - 4000 k n - 10820 k n - 5 3 6 3 7 3 > 25821 k n - 26543 k n - 10380 k n - 8 3 9 3 10 3 > 623 k n + 41 k n - 90 k n + 11 3 4 4 > 12 k n + 5766 n - 3530 k n - 2 4 3 4 4 4 > 6768 k n + 10614 k n + 11052 k n - 5 4 6 4 7 4 > 10104 k n - 23623 k n - 10791 k n + 8 4 9 4 10 4 > 1188 k n + 934 k n + 4 k n + 11 4 5 5 > 16 k n + 16843 n - 2858 k n + 2 5 3 5 4 5 > 746 k n + 38163 k n + 42580 k n + 5 5 6 5 7 5 > 16154 k n - 10533 k n - 9766 k n + 8 5 9 5 10 5 > 632 k n + 772 k n - 32 k n + 6 6 2 6 > 29527 n - 7491 k n + 7678 k n + 3 6 4 6 5 6 > 53925 k n + 48199 k n + 23181 k n + 6 6 7 6 8 6 > 144 k n - 6896 k n - 312 k n + 9 6 7 7 > 384 k n + 31224 n - 28137 k n + 2 7 3 7 4 7 > 3391 k n + 47796 k n + 29121 k n + 5 7 6 7 7 7 > 11862 k n + 3472 k n - 3104 k n - 8 7 9 7 8 > 480 k n + 64 k n + 18210 n - 8 2 8 3 8 > 55492 k n - 8925 k n + 31261 k n + 4 8 5 8 6 8 > 13332 k n + 2352 k n + 2400 k n - 7 8 8 8 9 > 592 k n - 128 k n + 3203 n - 9 2 9 3 9 > 62326 k n - 16843 k n + 13243 k n + 4 9 5 9 6 9 > 7044 k n + 160 k n + 704 k n - 10 10 2 10 > 2677 n - 41679 k n - 13783 k n + 3 10 4 10 5 10 > 526 k n + 3008 k n + 176 k n + 6 10 11 11 > 64 k n - 1790 n - 15791 k n - 2 11 3 11 4 11 > 5478 k n - 2604 k n + 544 k n + 5 11 12 12 > 64 k n - 314 n - 2582 k n - 2 12 3 12 13 > 516 k n - 1088 k n + 44 n + 13 2 13 3 13 > 156 k n + 288 k n - 128 k n + 14 14 2 14 2 > 16 n + 80 k n + 64 k n ) S S + n k 2 3 4 5 > (32 k + 198 k + 556 k + 968 k + 6 7 8 9 > 1168 k + 992 k + 572 k + 210 k + 10 11 > 44 k + 4 k + 16 n + 82 k n + 2 3 4 > 377 k n + 1479 k n + 3655 k n + 5 6 7 > 5830 k n + 6389 k n + 4893 k n + 8 9 10 > 2539 k n + 834 k n + 154 k n + 11 2 2 > 12 k n + 110 n + 482 k n + 2 2 3 2 4 2 > 1425 k n + 3853 k n + 8066 k n + 5 2 6 2 7 2 > 11914 k n + 12282 k n + 8731 k n + 8 2 9 2 10 2 > 4103 k n + 1168 k n + 170 k n + 11 2 3 3 > 8 k n + 275 n + 673 k n + 2 3 3 3 4 3 > 1470 k n + 2787 k n + 4725 k n + 5 3 6 3 7 3 > 7466 k n + 8612 k n + 6482 k n + 8 3 9 3 10 3 > 3060 k n + 816 k n + 92 k n + 4 4 2 4 > 327 n - 905 k n - 2405 k n - 3 4 4 4 5 4 > 5015 k n - 8044 k n - 6028 k n - 6 4 7 4 8 4 > 1553 k n + 700 k n + 780 k n + 9 4 10 4 5 > 272 k n + 32 k n + 283 n - 5 2 5 3 5 > 3123 k n - 5310 k n - 9589 k n - 4 5 5 5 6 5 > 14887 k n - 11654 k n - 5248 k n - 7 5 8 5 6 > 1544 k n - 224 k n + 337 n - 6 2 6 3 6 > 2795 k n - 34 k n - 1473 k n - 4 6 5 6 6 6 > 7177 k n - 5684 k n - 2280 k n - 7 6 8 6 7 > 624 k n - 80 k n + 268 n - 7 2 7 3 7 > 1270 k n + 5854 k n + 6674 k n + 4 7 5 7 6 7 > 1432 k n + 24 k n - 56 k n - 7 7 8 8 > 32 k n - 116 n - 1296 k n + 2 8 3 8 4 8 > 4121 k n + 4480 k n + 1732 k n + 5 8 6 8 9 > 560 k n + 80 k n - 342 n - 9 2 9 3 9 > 1472 k n + 510 k n + 476 k n + 4 9 5 9 10 > 160 k n + 64 k n - 146 n - 10 2 10 3 10 > 604 k n - 226 k n - 248 k n - 4 10 11 11 > 48 k n + 28 n + 24 k n + 2 11 3 11 12 > 28 k n - 32 k n + 16 n + 12 2 12 2 > 32 k n + 16 k n ) S + n 2 3 4 > (192 k + 812 k + 1542 k + 1644 k - 5 6 7 8 > 688 k - 5324 k - 7020 k - 4266 k - 9 10 11 12 > 1308 k - 214 k - 42 k - 14 k - 13 2 > 2 k + 1608 k n + 6316 k n + 3 4 5 > 9607 k n + 7773 k n - 4118 k n - 6 7 8 > 29055 k n - 39966 k n - 25111 k n - 9 10 11 > 7912 k n - 1283 k n - 199 k n - 12 13 2 2 > 58 k n - 8 k n - 468 n + 6158 k n + 2 2 3 2 4 2 > 21773 k n + 22040 k n + 10692 k n - 5 2 6 2 7 2 > 670 k n - 48559 k n - 84990 k n - 8 2 9 2 10 2 > 59722 k n - 19898 k n - 3092 k n - 11 2 12 2 13 2 > 300 k n - 66 k n - 8 k n - 3 3 2 3 > 5744 n + 14921 k n + 51396 k n + 3 3 4 3 5 3 > 23461 k n - 7839 k n + 32554 k n + 6 3 7 3 8 3 > 5643 k n - 72938 k n - 72758 k n - 9 3 10 3 11 3 > 27278 k n - 4092 k n - 202 k n - 12 3 4 4 > 20 k n - 29873 n + 23524 k n + 2 4 3 4 4 4 > 95303 k n + 8417 k n - 50483 k n + 5 4 6 4 > 71622 k n + 116632 k n + 7 4 8 4 9 4 > 11613 k n - 43235 k n - 21702 k n - 10 4 11 4 12 4 > 3272 k n - 108 k n - 16 k n - 5 5 2 5 > 86205 n + 24057 k n + 115026 k n - 3 5 4 5 > 42352 k n - 102358 k n + 5 5 6 5 > 54435 k n + 158506 k n + 7 5 8 5 9 5 > 78105 k n - 3752 k n - 9832 k n - 10 5 11 5 6 > 1504 k n + 16 k n - 151391 n + 6 2 6 3 6 > 40221 k n + 60095 k n - 170474 k n - 4 6 5 6 > 156743 k n - 13379 k n + 6 6 7 6 8 6 > 93985 k n + 69480 k n + 11144 k n - 9 6 10 6 7 > 2304 k n - 480 k n - 164600 n + 7 2 7 3 7 > 132352 k n - 4143 k n - 302530 k n - 4 7 5 7 > 186286 k n - 60163 k n + 6 7 7 7 8 7 > 16244 k n + 28784 k n + 7096 k n - 9 7 10 7 8 > 48 k n - 64 k n - 102175 n + 8 2 8 > 284190 k n + 33158 k n - 3 8 4 8 > 285399 k n - 150401 k n - 5 8 6 8 7 8 > 53186 k n - 10740 k n + 5096 k n + 8 8 9 8 9 > 1744 k n + 64 k n - 20143 n + 9 2 9 > 357177 k n + 118221 k n - 3 9 4 9 > 139050 k n - 74186 k n - 5 9 6 9 7 9 > 24868 k n - 6784 k n - 16 k n + 8 9 10 10 > 128 k n + 17985 n + 272241 k n + 2 10 3 10 > 125459 k n - 20100 k n - 4 10 5 10 > 18950 k n - 6824 k n - 6 10 7 10 11 > 1488 k n - 64 k n + 13472 n + 11 2 11 > 123433 k n + 64016 k n + 3 11 4 11 5 11 > 12570 k n - 708 k n - 1040 k n - 6 11 12 12 > 128 k n + 1938 n + 28882 k n + 2 12 3 12 4 12 > 14876 k n + 6820 k n + 768 k n - 5 12 13 13 > 64 k n - 1244 n + 1108 k n + 2 13 3 13 4 13 > 192 k n + 1216 k n + 128 k n - 14 14 2 14 > 544 n - 880 k n - 528 k n + 3 14 15 15 2 15 > 64 k n - 64 n - 128 k n - 64 k n 2 3 4 > ) S S + (192 k + 1036 k + 2466 k + n k 5 6 7 8 > 3918 k + 5222 k + 5358 k + 3490 k + 9 10 11 12 > 1266 k + 234 k + 36 k + 12 k + 13 2 3 > 2 k - 440 k n - 36 k n + 7286 k n + 4 5 6 > 23568 k n + 39806 k n + 49862 k n + 7 8 9 > 47574 k n + 29488 k n + 10356 k n + 10 11 12 > 1812 k n + 180 k n + 38 k n + 13 2 2 > 6 k n + 248 n - 3492 k n - 2 2 3 2 4 2 > 9954 k n + 13844 k n + 89052 k n + 5 2 6 2 > 162188 k n + 193972 k n + 7 2 8 2 > 176488 k n + 104710 k n + 9 2 10 2 11 2 > 35130 k n + 5770 k n + 380 k n + 12 2 13 2 3 > 26 k n + 4 k n + 2492 n - 3 2 3 3 3 > 9758 k n - 52172 k n - 26316 k n + 4 3 5 3 > 158086 k n + 333222 k n + 6 3 7 3 > 385310 k n + 351708 k n + 8 3 9 3 > 204840 k n + 65132 k n + 10 3 11 3 12 3 > 9958 k n + 480 k n - 16 k n + 4 4 2 4 > 9960 n - 8058 k n - 130770 k n - 3 4 4 4 > 179104 k n + 82812 k n + 5 4 6 4 > 334466 k n + 372276 k n + 7 4 8 4 > 393388 k n + 241668 k n + 9 4 10 4 11 4 > 73046 k n + 10390 k n + 404 k n - 12 4 5 5 > 32 k n + 18900 n + 11294 k n - 2 5 3 5 > 196272 k n - 406312 k n - 4 5 5 5 > 186318 k n + 67150 k n + 6 5 7 5 > 33142 k n + 209768 k n + 8 5 9 5 > 173844 k n + 50834 k n + 10 5 11 5 12 5 > 6832 k n + 240 k n - 16 k n + 6 6 2 6 > 11712 n + 19558 k n - 189890 k n - 3 6 4 6 > 526634 k n - 421186 k n - 5 6 6 6 > 170310 k n - 326676 k n - 7 6 8 6 9 6 > 25544 k n + 70752 k n + 20848 k n + 10 6 11 6 7 > 2736 k n + 80 k n - 19676 n - 7 2 7 > 16884 k n - 108500 k n - 3 7 4 7 > 400116 k n - 392508 k n - 5 7 6 7 > 97172 k n - 374356 k n - 7 7 8 7 9 7 > 113016 k n + 11856 k n + 4080 k n + 10 7 8 8 > 592 k n - 47984 n - 61634 k n - 2 8 3 8 > 3902 k n - 113604 k n - 4 8 5 8 > 198848 k n + 115974 k n - 6 8 7 8 8 8 > 194148 k n - 73432 k n - 944 k n + 9 8 10 8 9 > 48 k n + 64 k n - 43932 n - 9 2 9 3 9 > 63482 k n + 46674 k n + 113164 k n - 4 9 5 9 > 72218 k n + 180578 k n - 6 9 7 9 8 9 > 43856 k n - 23392 k n - 352 k n - 9 9 10 10 > 64 k n - 17748 n - 45248 k n + 2 10 3 10 > 4578 k n + 174588 k n - 4 10 5 10 > 47702 k n + 101412 k n + 6 10 7 10 11 > 1920 k n - 3872 k n + 6648 n - 11 2 11 > 39872 k n - 72194 k n + 3 11 4 11 > 128984 k n - 36064 k n + 5 11 6 11 7 11 > 26528 k n + 2848 k n - 256 k n + 12 12 2 12 > 22296 n - 30478 k n - 98226 k n + 3 12 4 12 > 63876 k n - 14176 k n + 5 12 6 12 13 > 2336 k n + 384 k n + 26960 n - 13 2 13 > 8506 k n - 65952 k n + 3 13 4 13 5 13 > 20848 k n - 2128 k n - 128 k n + 14 14 2 14 > 19548 n + 5672 k n - 25264 k n + 3 14 15 15 > 3792 k n + 8416 n + 5488 k n - 2 15 3 15 16 > 5232 k n + 256 k n + 1968 n + 16 2 16 17 > 1712 k n - 448 k n + 192 n + 17 2 > 192 k n ) S + k 2 3 > (-192 - 1580 k - 6758 k - 18396 k - 4 5 6 > 33566 k - 41868 k - 35790 k - 7 8 9 10 > 20562 k - 7422 k - 1288 k + 134 k + 11 12 13 > 124 k + 26 k + 2 k - 1908 n - 2 3 > 13972 k n - 54118 k n - 138366 k n - 4 5 6 > 245217 k n - 302303 k n - 256331 k n - 7 8 9 > 145912 k n - 52713 k n - 10181 k n - 10 11 12 > 133 k n + 382 k n + 72 k n + 13 2 2 > 4 k n - 7382 n - 49072 k n - 2 2 3 2 > 170912 k n - 403258 k n - 4 2 5 2 > 698734 k n - 872906 k n - 6 2 7 2 > 752921 k n - 431900 k n - 8 2 9 2 > 156543 k n - 31620 k n - 10 2 11 2 12 2 > 1936 k n + 428 k n + 64 k n - 3 3 2 3 > 11942 n - 78890 k n - 251190 k n - 3 3 4 3 > 522652 k n - 908892 k n - 5 3 6 3 > 1240476 k n - 1152875 k n - 7 3 8 3 > 687377 k n - 252651 k n - 9 3 10 3 11 3 > 51817 k n - 3928 k n + 358 k n + 12 3 4 4 > 56 k n + 3702 n - 22454 k n - 2 4 3 4 > 99432 k n - 87486 k n - 4 4 5 4 > 250432 k n - 727775 k n - 6 4 7 4 > 923468 k n - 620577 k n - 8 4 9 4 > 237837 k n - 49720 k n - 10 4 11 4 12 4 > 4350 k n + 68 k n + 16 k n + 5 5 2 5 > 51659 n + 127951 k n + 201835 k n + 3 5 4 5 > 651806 k n + 882718 k n + 5 5 6 5 > 319144 k n - 241913 k n - 7 5 8 5 > 293183 k n - 128584 k n - 9 5 10 5 11 5 > 27348 k n - 2364 k n - 16 k n + 6 6 2 6 > 101864 n + 209718 k n + 235413 k n + 3 6 4 6 > 885748 k n + 1367191 k n + 5 6 6 6 > 866439 k n + 215559 k n - 7 6 8 6 9 6 > 32252 k n - 35524 k n - 8288 k n - 10 6 7 7 > 608 k n + 107760 n + 93934 k n - 2 7 3 7 > 136475 k n + 405493 k n + 4 7 5 7 > 913009 k n + 639713 k n + 6 7 7 7 8 7 > 228992 k n + 36600 k n - 2192 k n - 9 7 10 7 8 > 1232 k n - 64 k n + 70770 n - 8 2 8 > 95257 k n - 494742 k n - 3 8 4 8 > 135983 k n + 265009 k n + 5 8 6 8 > 217906 k n + 88464 k n + 7 8 8 8 9 8 > 18728 k n + 1104 k n - 64 k n + 9 9 2 9 > 32457 n - 162551 k n - 488122 k n - 3 9 4 9 > 259887 k n - 20380 k n + 5 9 6 9 7 9 > 18056 k n + 13608 k n + 3408 k n + 8 9 10 10 > 192 k n + 13908 n - 104719 k n - 2 10 3 10 > 257191 k n - 130616 k n - 4 10 5 10 6 10 > 35948 k n - 9952 k n - 208 k n + 7 10 11 11 > 192 k n + 6738 n - 35784 k n - 2 11 3 11 > 76992 k n - 27582 k n - 4 11 5 11 6 11 > 8264 k n - 2800 k n - 192 k n + 12 12 2 12 > 2610 n - 6516 k n - 12242 k n - 3 12 4 12 5 12 > 524 k n - 288 k n - 192 k n + 13 13 2 13 > 564 n - 600 k n - 844 k n + 3 13 4 13 14 > 640 k n + 64 k n + 48 n - 14 2 14 3 14 > 32 k n - 16 k n + 64 k n ) S + n 2 3 4 > (192 k + 1580 k + 5222 k + 10404 k + 5 6 7 > 15262 k + 16146 k + 10034 k + 8 9 10 11 > 2438 k - 658 k - 604 k - 214 k - 12 13 14 > 70 k - 18 k - 2 k - 192 n + 2 3 > 808 k n + 15270 k n + 58034 k n + 4 5 6 > 114928 k n + 155414 k n + 155246 k n + 7 8 9 > 94672 k n + 22924 k n - 6050 k n - 10 11 12 > 5212 k n - 1394 k n - 274 k n - 13 14 2 > 56 k n - 6 k n - 2388 n - 2 2 2 3 2 > 8062 k n + 46690 k n + 256822 k n + 4 2 5 2 > 533514 k n + 683098 k n + 6 2 7 2 > 652520 k n + 394286 k n + 8 2 9 2 > 99230 k n - 20358 k n - 10 2 11 2 12 2 > 18546 k n - 4134 k n - 404 k n - 13 2 14 2 3 > 38 k n - 4 k n - 12430 n - 3 2 3 3 3 > 80882 k n - 19010 k n + 535082 k n + 4 3 5 3 > 1315160 k n + 1662734 k n + 6 3 7 3 > 1567560 k n + 958940 k n + 8 3 9 3 > 259980 k n - 30254 k n - 10 3 11 3 12 3 > 35642 k n - 7324 k n - 380 k n + 13 3 4 4 > 16 k n - 34246 n - 318756 k n - 2 4 3 4 > 501188 k n + 305402 k n + 4 4 5 4 > 1694688 k n + 2323390 k n + 6 4 7 4 > 2327288 k n + 1504848 k n + 8 4 9 4 > 451626 k n - 10730 k n - 10 4 11 4 12 4 > 40466 k n - 8226 k n - 292 k n + 13 4 5 5 > 32 k n - 49246 n - 719600 k n - 2 5 3 5 > 1488602 k n - 1000102 k n + 4 5 5 5 > 546836 k n + 1514012 k n + 6 5 7 5 > 2079190 k n + 1565828 k n + 8 5 9 5 > 533994 k n + 30434 k n - 10 5 11 5 12 5 > 27146 k n - 5904 k n - 192 k n + 13 5 6 6 > 16 k n - 19368 n - 1026502 k n - 2 6 3 6 > 2175538 k n - 2537186 k n - 4 6 5 6 > 1712016 k n - 551192 k n + 6 6 7 6 > 837346 k n + 1044216 k n + 8 6 9 6 > 424748 k n + 51060 k n - 10 6 11 6 12 6 > 9440 k n - 2512 k n - 80 k n + 7 7 2 7 > 46164 n - 981504 k n - 1477228 k n - 3 7 4 7 > 2466838 k n - 2955882 k n - 5 7 6 7 > 2066274 k n - 344176 k n + 7 7 8 7 > 377980 k n + 217728 k n + 9 7 10 7 11 7 > 36240 k n - 848 k n - 528 k n + 8 8 2 8 > 50092 n - 754672 k n + 308454 k n - 3 8 4 8 > 581646 k n - 2130314 k n - 5 8 6 8 > 1911042 k n - 679526 k n + 7 8 8 8 9 8 > 3492 k n + 65736 k n + 13520 k n + 10 8 11 8 9 > 272 k n - 64 k n - 85252 n - 9 2 9 > 739440 k n + 1578866 k n + 3 9 4 9 > 1170306 k n - 560914 k n - 5 9 6 9 > 870232 k n - 413474 k n - 7 9 8 9 9 9 > 65088 k n + 8928 k n + 2720 k n + 10 9 10 10 > 64 k n - 282976 n - 930158 k n + 2 10 3 10 > 1489772 k n + 1410366 k n + 4 10 5 10 > 231628 k n - 135894 k n - 6 10 7 10 > 124308 k n - 30016 k n - 8 10 9 10 11 > 480 k n + 256 k n - 378222 n - 11 2 11 > 956740 k n + 728266 k n + 3 11 4 11 > 732176 k n + 226220 k n + 5 11 6 11 > 52560 k n - 13632 k n - 7 11 8 11 12 > 5856 k n - 256 k n - 311286 n - 12 2 12 > 678930 k n + 185674 k n + 3 12 4 12 > 182718 k n + 62076 k n + 5 12 6 12 7 12 > 29344 k n + 1632 k n - 384 k n - 13 13 > 170982 n - 321602 k n + 2 13 3 13 > 17682 k n + 7760 k n + 4 13 5 13 6 13 > 1888 k n + 5008 k n + 384 k n - 14 14 2 14 > 63028 n - 98744 k n + 1144 k n - 3 14 4 14 5 14 > 6512 k n - 2000 k n + 256 k n - 15 15 2 15 > 14992 n - 18416 k n + 1936 k n - 3 15 4 15 16 > 1296 k n - 256 k n - 2080 n - 16 2 16 3 16 > 1808 k n + 656 k n - 64 k n - 17 17 2 17 > 128 n - 64 k n + 64 k n ) S + k 2 3 > (384 + 2584 k + 8584 k + 18010 k + 4 5 6 7 > 24822 k + 20608 k + 6136 k - 6372 k - 8 9 10 11 > 8712 k - 4896 k - 1486 k - 190 k + 12 13 14 > 30 k + 16 k + 2 k + 4776 n + 2 3 > 31704 k n + 101770 k n + 203906 k n + 4 5 6 > 272938 k n + 230466 k n + 86300 k n - 7 8 9 > 40960 k n - 70398 k n - 39904 k n - 10 11 12 > 11776 k n - 1574 k n + 80 k n + 13 14 2 > 58 k n + 6 k n + 25216 n + 2 2 2 > 166926 k n + 520414 k n + 3 2 4 2 > 996734 k n + 1300100 k n + 5 2 6 2 > 1122420 k n + 500828 k n - 7 2 8 2 > 69060 k n - 231822 k n - 9 2 10 2 > 136692 k n - 39502 k n - 11 2 12 2 13 2 > 5194 k n + 50 k n + 82 k n + 14 2 3 3 > 4 k n + 72286 n + 489766 k n + 2 3 3 3 > 1492982 k n + 2728022 k n + 4 3 5 3 > 3489390 k n + 3118544 k n + 6 3 7 3 > 1607180 k n + 125128 k n - 8 3 9 3 > 390234 k n - 255344 k n - 10 3 11 3 13 3 > 73672 k n - 9384 k n + 88 k n + 4 4 2 4 > 115284 n + 862522 k n + 2596238 k n + 3 4 4 4 > 4469360 k n + 5676252 k n + 5 4 6 4 > 5409020 k n + 3183978 k n + 7 4 8 4 > 709352 k n - 320512 k n - 9 4 10 4 > 279616 k n - 83446 k n - 11 4 12 4 13 4 > 10574 k n - 112 k n + 64 k n + 5 5 2 5 > 75960 n + 895976 k n + 2697572 k n + 3 5 4 5 > 4115842 k n + 5365900 k n + 5 5 6 5 > 5936006 k n + 4068670 k n + 7 5 8 5 > 1327626 k n - 26670 k n - 9 5 10 5 > 174928 k n - 57942 k n - 11 5 12 5 13 5 > 7368 k n - 176 k n + 16 k n - 6 6 2 6 > 65322 n + 482416 k n + 1415058 k n + 3 6 4 6 > 1074696 k n + 1880630 k n + 5 6 6 6 > 3785154 k n + 3343248 k n + 7 6 8 6 > 1379758 k n + 198752 k n - 9 6 10 6 > 49628 k n - 23644 k n - 11 6 12 6 7 > 2896 k n - 64 k n - 182062 n + 7 2 7 > 163454 k n + 79036 k n - 3 7 4 7 > 2173338 k n - 1994974 k n + 5 7 6 7 > 727080 k n + 1631064 k n + 7 7 8 7 > 856680 k n + 195808 k n + 9 7 10 7 11 7 > 5808 k n - 5200 k n - 592 k n - 8 8 2 8 > 134978 n + 441624 k n + 53134 k n - 3 8 4 8 > 2985398 k n - 3286000 k n - 5 8 6 8 > 930002 k n + 300782 k n + 7 8 8 8 9 8 > 299020 k n + 90064 k n + 8672 k n - 10 8 11 8 9 > 512 k n - 64 k n + 33954 n + 9 2 9 > 1002108 k n + 885498 k n - 3 9 4 9 > 1698368 k n - 2184248 k n - 5 9 6 9 > 903894 k n - 138918 k n + 7 9 8 9 9 9 > 38176 k n + 21440 k n + 2464 k n + 10 10 > 160126 n + 1179838 k n + 2 10 3 10 > 1279122 k n - 348148 k n - 4 10 5 10 > 752686 k n - 357470 k n - 6 10 7 10 > 104932 k n - 9664 k n + 8 10 9 10 11 > 2176 k n + 256 k n + 164352 n + 11 2 11 > 853860 k n + 921704 k n + 3 11 4 11 > 120384 k n - 89200 k n - 5 11 6 11 > 55832 k n - 26816 k n - 7 11 12 12 > 4000 k n + 99390 n + 406290 k n + 2 12 3 12 > 392774 k n + 86946 k n + 4 12 5 12 > 25696 k n + 5280 k n - 6 12 7 12 13 > 2560 k n - 384 k n + 39838 n + 13 2 13 > 128748 k n + 99798 k n + 3 13 4 13 > 14824 k n + 9968 k n + 5 13 14 14 > 2896 k n + 10588 n + 26504 k n + 2 14 3 14 4 14 > 13820 k n - 1648 k n + 960 k n + 5 14 15 15 > 256 k n + 1712 n + 3280 k n + 2 15 3 15 16 > 784 k n - 784 k n + 128 n + 16 3 16 > 192 k n - 64 k n ), 2 3 4 5 > (-32 k - 268 k - 570 k - 670 k - 6 7 8 9 > 736 k - 648 k - 300 k - 48 k + 10 11 > 6 k + 2 k - 88 n - 348 k n - 2 3 4 > 1002 k n - 3419 k n - 6156 k n - 5 6 7 > 6102 k n - 4485 k n - 2867 k n - 8 9 10 > 1148 k n - 158 k n + 29 k n + 11 2 2 > 8 k n - 1092 n - 3372 k n - 2 2 3 2 4 2 > 6911 k n - 15901 k n - 24273 k n - 5 2 6 2 7 2 > 21145 k n - 10960 k n - 4218 k n - 8 2 9 2 10 2 > 1304 k n - 156 k n + 42 k n + 11 2 3 3 > 8 k n - 5766 n - 13313 k n - 2 3 3 3 4 3 > 19901 k n - 35093 k n - 46966 k n - 5 3 6 3 7 3 > 37503 k n - 14598 k n - 1927 k n - 8 3 9 3 10 3 > 115 k n - 48 k n + 20 k n - 4 4 2 4 > 16843 n - 26669 k n - 24479 k n - 3 4 4 4 5 4 > 35914 k n - 47607 k n - 38221 k n - 6 4 7 4 8 4 > 12718 k n + 1073 k n + 886 k n + 9 4 10 4 5 > 24 k n + 16 k n - 29527 n - 5 2 5 3 5 > 23733 k n + 2249 k n - 5027 k n - 4 5 5 5 6 5 > 22067 k n - 23251 k n - 8693 k n + 7 5 8 5 9 5 > 1518 k n + 796 k n - 16 k n - 6 6 2 6 > 31224 n + 9927 k n + 48898 k n + 3 6 4 6 5 6 > 26132 k n - 70 k n - 8549 k n - 6 6 7 6 8 6 > 5378 k n + 484 k n + 368 k n - 7 7 2 7 > 18210 n + 52289 k n + 73928 k n + 3 7 4 7 5 7 > 29051 k n + 3313 k n - 1906 k n - 6 7 7 7 8 7 > 2620 k n - 112 k n + 64 k n - 8 8 2 8 > 3203 n + 65003 k n + 60312 k n + 3 8 4 8 5 8 > 16645 k n + 446 k n - 220 k n - 6 8 7 8 9 > 704 k n - 64 k n + 2677 n + 9 2 9 3 9 > 43469 k n + 29888 k n + 7490 k n - 4 9 6 9 10 > 60 k n - 64 k n + 1790 n + 10 2 10 3 10 > 16105 k n + 8016 k n + 2948 k n + 4 10 11 11 > 176 k n + 314 n + 2538 k n + 2 11 3 11 4 11 > 344 k n + 720 k n + 64 k n - 12 12 2 12 > 44 n - 172 k n - 368 k n + 3 12 13 13 2 13 > 64 k n - 16 n - 80 k n - 64 k n ) 2 2 3 4 > S S + (392 k + 1858 k + 4272 k + n k 5 6 7 8 > 6574 k + 7212 k + 5340 k + 2452 k + 9 10 11 > 626 k + 72 k + 2 k + 268 n + 2 3 > 1272 k n + 5847 k n + 18903 k n + 4 5 6 > 38576 k n + 53009 k n + 50504 k n + 7 8 9 > 32439 k n + 13104 k n + 2969 k n + 10 11 2 > 301 k n + 8 k n + 2660 n + 2 2 2 3 2 > 11434 k n + 31029 k n + 72803 k n + 4 2 5 2 > 132867 k n + 168784 k n + 6 2 7 2 > 143889 k n + 80377 k n + 8 2 9 2 10 2 > 27811 k n + 5256 k n + 406 k n + 11 2 3 3 > 8 k n + 11109 n + 41051 k n + 2 3 3 3 > 74990 k n + 125863 k n + 4 3 5 3 > 214812 k n + 269064 k n + 6 3 7 3 > 214903 k n + 107050 k n + 8 3 9 3 10 3 > 31784 k n + 4774 k n + 204 k n + 4 4 2 4 > 25669 n + 74661 k n + 70421 k n + 3 4 4 4 > 51213 k n + 129847 k n + 5 4 6 4 > 213488 k n + 176315 k n + 7 4 8 4 9 4 > 81958 k n + 21712 k n + 2692 k n + 10 4 5 5 > 48 k n + 36460 n + 68378 k n - 2 5 3 5 > 48455 k n - 176499 k n - 4 5 5 5 6 5 > 94920 k n + 45502 k n + 70504 k n + 7 5 8 5 9 5 > 33592 k n + 8248 k n + 864 k n + 6 6 2 6 > 34258 n + 18361 k n - 194711 k n - 3 6 4 6 > 349723 k n - 227499 k n - 5 6 6 6 7 6 > 56990 k n + 3992 k n + 5736 k n + 8 6 9 6 7 > 1472 k n + 128 k n + 23789 n - 7 2 7 > 20068 k n - 215897 k n - 3 7 4 7 > 301670 k n - 173834 k n - 5 7 6 7 7 7 > 51100 k n - 7168 k n - 208 k n + 8 7 8 8 > 64 k n + 13755 n - 19992 k n - 2 8 3 8 > 123335 k n - 138864 k n - 4 8 5 8 6 8 > 64188 k n - 17248 k n - 2464 k n - 7 8 9 9 > 128 k n + 5906 n - 8363 k n - 2 9 3 9 4 9 > 39173 k n - 32096 k n - 9160 k n - 5 9 6 9 10 > 2224 k n - 256 k n + 828 n - 10 2 10 3 10 > 3458 k n - 7470 k n - 2608 k n + 4 10 11 11 > 576 k n - 654 n - 1706 k n - 2 11 3 11 4 11 > 1164 k n + 80 k n + 192 k n - 12 12 2 12 > 340 n - 484 k n - 144 k n - 13 13 2 > 48 n - 48 k n ) S + n 2 3 4 > (2160 k + 5992 k + 15856 k + 24952 k + 5 6 7 > 7672 k - 19240 k - 22144 k - 8 9 10 11 > 10736 k - 3320 k - 944 k - 224 k - 12 > 24 k + 4224 n + 34800 k n + 2 3 4 > 77484 k n + 180216 k n + 268912 k n + 5 6 7 > 99802 k n - 158732 k n - 190194 k n - 8 9 10 > 87860 k n - 23374 k n - 5432 k n - 11 12 2 > 1194 k n - 124 k n + 57568 n + 2 2 2 > 247060 k n + 436360 k n + 3 2 4 2 > 889870 k n + 1281716 k n + 5 2 6 2 > 571317 k n - 525252 k n - 7 2 8 2 > 692911 k n - 306392 k n - 9 2 10 2 > 68647 k n - 11780 k n - 11 2 12 2 3 > 2245 k n - 216 k n + 342448 n + 3 2 3 > 999356 k n + 1389488 k n + 3 3 4 3 > 2504347 k n + 3565853 k n + 5 3 6 3 > 1886717 k n - 836906 k n - 7 3 8 3 > 1399523 k n - 604773 k n - 9 3 10 3 > 114729 k n - 12998 k n - 11 3 12 3 4 > 1908 k n - 144 k n + 1168540 n + 4 2 4 > 2476810 k n + 2647777 k n + 3 4 4 4 > 4414862 k n + 6439816 k n + 5 4 6 4 > 3969887 k n - 454055 k n - 7 4 8 4 > 1706392 k n - 749876 k n - 9 4 10 4 11 4 > 124321 k n - 8918 k n - 944 k n - 12 4 5 5 > 32 k n + 2521800 n + 3646091 k n + 2 5 3 5 > 2628514 k n + 4958212 k n + 4 5 5 5 > 7955570 k n + 5578465 k n + 6 5 7 5 > 613670 k n - 1262878 k n - 8 5 9 5 > 605648 k n - 89446 k n - 10 5 11 5 6 > 3716 k n - 336 k n + 3556347 n + 6 2 6 > 2251005 k n - 449458 k n + 3 6 4 6 > 3165599 k n + 6904361 k n + 5 6 6 6 > 5366804 k n + 1405964 k n - 7 6 8 6 > 513944 k n - 322284 k n - 9 6 10 6 11 6 > 43060 k n - 672 k n - 64 k n + 7 7 > 3215093 n - 2471183 k n - 2 7 3 7 > 5830314 k n + 114827 k n + 4 7 5 7 > 4213622 k n + 3571128 k n + 6 7 7 7 > 1305710 k n - 47728 k n - 8 7 9 7 8 > 111104 k n - 13952 k n + 1638878 n - 8 2 8 > 7526735 k n - 9776893 k n - 3 8 4 8 > 2146504 k n + 1694780 k n + 5 8 6 8 > 1627988 k n + 715964 k n + 7 8 8 8 9 8 > 60224 k n - 22800 k n - 2816 k n + 9 9 > 147438 n - 8952643 k n - 2 9 3 9 > 9398734 k n - 2580368 k n + 4 9 5 9 > 293946 k n + 483404 k n + 6 9 7 9 8 9 > 245432 k n + 31824 k n - 2112 k n - 9 9 10 10 > 256 k n - 376597 n - 6429037 k n - 2 10 3 10 > 5842024 k n - 1738627 k n - 4 10 5 10 > 131096 k n + 78184 k n + 6 10 7 10 11 > 50928 k n + 6656 k n - 226865 n - 11 2 11 > 2936407 k n - 2334904 k n - 3 11 4 11 > 751788 k n - 117312 k n + 5 11 6 11 7 11 > 688 k n + 5696 k n + 512 k n - 12 12 > 23024 n - 799699 k n - 2 12 3 12 > 537750 k n - 205840 k n - 4 12 5 12 6 12 > 40592 k n - 2048 k n + 256 k n + 13 13 2 13 > 30242 n - 89814 k n - 37364 k n - 3 13 4 13 5 13 > 32448 k n - 7104 k n - 256 k n + 14 14 2 14 > 15628 n + 13644 k n + 13424 k n - 3 14 4 14 15 > 2240 k n - 512 k n + 3216 n + 15 2 15 16 > 5520 k n + 3520 k n + 256 n + 16 2 16 > 512 k n + 256 k n ) S S + n k 2 3 4 5 > (2160 k + 7368 k + 11536 k + 16536 k + 6 7 8 9 > 22856 k + 19740 k + 8796 k + 2208 k + 10 11 12 > 704 k + 228 k + 28 k - 3872 k n + 2 3 4 > 15088 k n + 73540 k n + 125580 k n + 5 6 7 > 190836 k n + 262276 k n + 218510 k n + 8 9 10 > 93142 k n + 20904 k n + 4920 k n + 11 12 2 > 1202 k n + 114 k n + 1712 n - 2 2 2 3 2 > 43576 k n + 6772 k n + 289620 k n + 4 2 5 2 > 560434 k n + 916984 k n + 6 2 7 2 > 1291004 k n + 1049288 k n + 8 2 9 2 > 424596 k n + 84920 k n + 10 2 11 2 12 2 > 15120 k n + 2556 k n + 154 k n + 3 3 2 3 > 21120 n - 206276 k n - 276592 k n + 3 3 4 3 > 475426 k n + 1232584 k n + 5 3 6 3 > 2311434 k n + 3558998 k n + 7 3 8 3 > 2891524 k n + 1106880 k n + 9 3 10 3 > 194926 k n + 26822 k n + 11 3 12 3 4 > 2862 k n + 84 k n + 114428 n - 4 2 4 > 517092 k n - 1367478 k n - 3 4 4 4 > 251968 k n + 954792 k n + 5 4 6 4 > 2938108 k n + 5959816 k n + 7 4 8 4 > 5086748 k n + 1847732 k n + 9 4 10 4 > 282120 k n + 30210 k n + 11 4 12 4 5 > 1600 k n + 16 k n + 361948 n - 5 2 5 > 635518 k n - 3436528 k n - 3 5 4 5 > 2819180 k n - 1778448 k n + 5 5 6 5 > 419792 k n + 5967252 k n + 7 5 8 5 > 5987166 k n + 2086810 k n + 9 5 10 5 > 271722 k n + 23124 k n + 11 5 6 6 > 176 k n + 753444 n + 149364 k n - 2 6 3 6 > 5385142 k n - 6406344 k n - 4 6 5 6 > 6002132 k n - 4890832 k n + 6 6 7 6 > 2794554 k n + 4777164 k n + 8 6 9 6 > 1631860 k n + 175824 k n + 10 6 11 6 7 > 12384 k n - 208 k n + 1107548 n + 7 2 7 > 2234988 k n - 5277510 k n - 3 7 4 7 > 8375308 k n - 8123712 k n - 5 7 6 7 > 8841178 k n - 961420 k n + 7 7 8 7 > 2543996 k n + 887320 k n + 9 7 10 7 11 7 > 73984 k n + 4192 k n - 64 k n + 8 8 > 1184798 n + 4983396 k n - 2 8 3 8 > 2279544 k n - 7048320 k n - 4 8 5 8 > 6281178 k n - 8355268 k n - 6 8 7 8 > 2508012 k n + 851360 k n + 8 8 9 8 10 8 > 333616 k n + 19152 k n + 640 k n + 9 9 > 837162 n + 6863808 k n + 2 9 3 9 > 2072644 k n - 3610360 k n - 4 9 5 9 > 2644410 k n - 4956924 k n - 6 9 7 9 > 1906952 k n + 143968 k n + 8 9 9 9 10 > 84992 k n + 2944 k n + 75840 n + 10 2 10 > 6551864 k n + 5009566 k n - 3 10 4 10 > 566664 k n - 180718 k n - 5 10 6 10 > 1889952 k n - 828160 k n - 7 10 8 10 9 10 > 5408 k n + 13568 k n + 256 k n - 11 11 > 812592 n + 4209940 k n + 2 11 3 11 > 5077382 k n + 754870 k n + 4 11 5 11 > 473396 k n - 436864 k n - 6 11 7 11 > 218560 k n - 7296 k n + 8 11 12 > 1024 k n - 1327518 n + 12 2 12 > 1487612 k n + 3239786 k n + 3 12 4 12 > 741344 k n + 299952 k n - 5 12 6 12 > 48224 k n - 32512 k n - 7 12 13 > 1024 k n - 1212818 n - 13 2 13 > 103118 k n + 1369524 k n + 3 13 4 13 > 348528 k n + 91520 k n + 5 13 6 13 14 > 768 k n - 2048 k n - 726688 n - 14 2 14 > 435696 k n + 373152 k n + 3 14 4 14 > 96496 k n + 14848 k n + 5 14 15 15 > 512 k n - 290976 n - 245504 k n + 2 15 3 15 > 59488 k n + 15040 k n + 4 15 16 16 > 1024 k n - 75248 n - 72048 k n + 2 16 3 16 17 > 4224 k n + 1024 k n - 11392 n - 17 18 18 2 > 11392 k n - 768 n - 768 k n ) S + k 2 3 > (-2160 - 13096 k - 46488 k - 116848 k - 4 5 6 > 200128 k - 226712 k - 167368 k - 7 8 9 10 > 77152 k - 18560 k + 272 k + 1520 k + 11 12 > 384 k + 32 k - 31184 n - 160996 k n - 2 3 > 496432 k n - 1165668 k n - 4 5 > 1931954 k n - 2117686 k n - 6 7 > 1499666 k n - 663714 k n - 8 9 10 > 162518 k n - 9134 k n + 5938 k n + 11 12 2 > 1486 k n + 104 k n - 202532 n - 2 2 2 > 873480 k n - 2250474 k n - 3 2 4 2 > 4890518 k n - 7947729 k n - 5 2 6 2 > 8536299 k n - 5835777 k n - 7 2 8 2 > 2469321 k n - 590263 k n - 9 2 10 2 11 2 > 50329 k n + 8891 k n + 2155 k n + 12 2 3 3 > 92 k n - 779608 n - 2735868 k n - 2 3 3 3 > 5494368 k n - 10921908 k n - 4 3 5 3 > 18014104 k n - 19434475 k n - 6 3 7 3 > 12983073 k n - 5265440 k n - 8 3 9 3 > 1203510 k n - 110655 k n + 10 3 11 3 12 3 > 8239 k n + 1946 k n + 24 k n - 4 4 > 1974006 n - 5377238 k n - 2 4 3 4 > 7050518 k n - 12591050 k n - 4 4 5 4 > 23752482 k n - 27296820 k n - 6 4 7 4 > 18217746 k n - 7139332 k n - 8 4 9 4 > 1557008 k n - 143980 k n + 10 4 11 4 5 > 4576 k n + 1236 k n - 3459522 n - 5 2 5 > 6576042 k n - 1559272 k n - 3 5 4 5 > 2037354 k n - 15960564 k n - 5 5 6 5 > 23867518 k n - 16694214 k n - 7 5 8 5 > 6401580 k n - 1329024 k n - 9 5 10 5 11 5 > 119822 k n + 728 k n + 448 k n - 6 6 > 4322309 n - 4126915 k n + 2 6 3 6 > 11164063 k n + 17113239 k n + 4 6 5 6 > 1224650 k n - 11701472 k n - 6 6 7 6 > 9918724 k n - 3825236 k n - 8 6 9 6 10 6 > 750012 k n - 62412 k n - 400 k n + 11 6 7 7 > 64 k n - 3941614 n + 856119 k n + 2 7 3 7 > 22667203 k n + 30166424 k n + 4 7 5 7 > 13479244 k n - 1125582 k n - 6 7 7 7 > 3549780 k n - 1498164 k n - 8 7 9 7 10 7 > 275520 k n - 19392 k n - 128 k n - 8 8 > 2704148 n + 4526018 k n + 2 8 3 8 > 24398786 k n + 28255812 k n + 4 8 5 8 > 13857366 k n + 2546918 k n - 6 8 7 8 > 527376 k n - 364424 k n - 8 8 9 8 9 > 63632 k n - 3328 k n - 1467036 n + 9 2 9 > 4742668 k n + 17151628 k n + 3 9 4 9 > 17045670 k n + 7812182 k n + 5 9 6 9 > 1870724 k n + 123424 k n - 7 9 8 9 9 9 > 47120 k n - 8512 k n - 256 k n - 10 10 > 672571 n + 2861001 k n + 2 10 3 10 > 8243593 k n + 6831897 k n + 4 10 5 10 > 2696492 k n + 651848 k n + 6 10 7 10 8 10 > 76400 k n - 1344 k n - 512 k n - 11 11 > 269246 n + 1106789 k n + 2 11 3 11 > 2698885 k n + 1756778 k n + 4 11 5 11 > 543000 k n + 123408 k n + 6 11 7 11 12 > 14592 k n + 256 k n - 88838 n + 12 2 12 > 273878 k n + 577440 k n + 3 12 4 12 > 252628 k n + 48016 k n + 5 12 6 12 13 > 11136 k n + 1024 k n - 21278 n + 13 2 13 > 40378 k n + 72936 k n + 3 13 4 13 5 13 > 9168 k n - 1856 k n + 256 k n - 14 14 2 14 > 3132 n + 2916 k n + 4128 k n - 3 14 4 14 15 > 2432 k n - 512 k n - 208 n + 15 3 15 > 48 k n - 256 k n ) S + n 2 3 4 > (2160 k + 13096 k + 27768 k + 46960 k + 5 6 7 > 72160 k + 63908 k + 20616 k - 8 9 10 11 > 5792 k - 6460 k - 2708 k - 1016 k - 12 13 > 264 k - 28 k - 2160 n + 19528 k n + 2 3 4 > 164508 k n + 356472 k n + 579268 k n + 5 6 7 > 852560 k n + 736130 k n + 234408 k n - 8 9 10 > 60368 k n - 63026 k n - 20242 k n - 11 12 13 > 5284 k n - 1152 k n - 114 k n - 2 2 2 2 > 32624 n + 24752 k n + 832392 k n + 3 2 4 2 > 1923602 k n + 3114122 k n + 5 2 6 2 > 4510940 k n + 3842848 k n + 7 2 8 2 > 1245346 k n - 246232 k n - 9 2 10 2 > 269222 k n - 71120 k n - 11 2 12 2 13 2 > 11656 k n - 1746 k n - 154 k n - 3 3 2 3 > 217028 n - 409928 k n + 1951342 k n + 3 3 4 3 > 5440954 k n + 9424854 k n + 5 3 6 3 > 14086670 k n + 12027244 k n + 7 3 8 3 > 4043388 k n - 493336 k n - 9 3 10 3 > 659018 k n - 154282 k n - 11 3 12 3 13 3 > 15846 k n - 1034 k n - 84 k n - 4 4 2 4 > 825896 n - 2614402 k n + 640746 k n + 3 4 4 4 > 7232620 k n + 16851494 k n + 5 4 6 4 > 28646524 k n + 25090216 k n + 7 4 8 4 > 8827422 k n - 423382 k n - 9 4 10 4 > 1021230 k n - 222774 k n - 11 4 12 4 13 4 > 15566 k n + 144 k n - 16 k n - 5 5 > 1911970 n - 7695034 k n - 2 5 3 5 > 9180250 k n - 3090794 k n + 4 5 5 5 > 15065362 k n + 39114404 k n + 6 5 7 5 > 36585990 k n + 13518474 k n + 8 5 9 5 > 204108 k n - 1046040 k n - 10 5 11 5 > 219662 k n - 11732 k n + 12 5 6 6 > 592 k n - 2417256 n - 12711776 k n - 2 6 3 6 > 29092598 k n - 32351496 k n - 4 6 5 6 > 4631234 k n + 34868262 k n + 6 6 7 6 > 37918894 k n + 14785222 k n + 8 6 9 6 > 950514 k n - 713132 k n - 10 6 11 6 > 147392 k n - 6640 k n + 12 6 7 7 > 336 k n - 33134 n - 9409428 k n - 2 7 3 7 > 47281006 k n - 68010164 k n - 4 7 5 7 > 33936298 k n + 17006446 k n + 6 7 7 7 > 27722274 k n + 11579228 k n + 8 7 9 7 > 1154352 k n - 316040 k n - 10 7 11 7 12 7 > 65056 k n - 2400 k n + 64 k n + 8 8 > 6583192 n + 7724334 k n - 2 8 3 8 > 46616396 k n - 84053926 k n - 4 8 5 8 > 51307870 k n - 1083242 k n + 6 8 7 8 > 13730944 k n + 6431396 k n + 8 8 9 8 > 787888 k n - 86688 k n - 10 8 11 8 9 > 17936 k n - 384 k n + 14990520 n + 9 2 9 > 31334502 k n - 25451812 k n - 3 9 4 9 > 69123342 k n - 45599946 k n - 5 9 6 9 > 8693978 k n + 4055908 k n + 7 9 8 9 > 2477320 k n + 334048 k n - 9 9 10 9 > 13632 k n - 2944 k n + 10 10 > 19987250 n + 45419552 k n - 2 10 3 10 > 1077558 k n - 38411526 k n - 4 10 5 10 > 26658866 k n - 7094274 k n + 6 10 7 10 > 307392 k n + 634144 k n + 8 10 9 10 > 88736 k n - 1024 k n - 10 10 11 > 256 k n + 18524626 n + 11 2 11 > 41706244 k n + 10581440 k n - 3 11 4 11 > 13654262 k n - 10243834 k n - 5 11 6 11 > 3191860 k n - 273600 k n + 7 11 8 11 > 98496 k n + 13952 k n + 12 12 > 12517780 n + 26752764 k n + 2 12 3 12 > 9493558 k n - 2419030 k n - 4 12 5 12 > 2357136 k n - 873616 k n - 6 12 7 12 > 120480 k n + 7168 k n + 8 12 13 > 1024 k n + 6237254 n + 13 2 13 > 12267748 k n + 4490442 k n + 3 13 4 13 > 199692 k n - 200752 k n - 5 13 6 13 > 136960 k n - 21760 k n + 14 14 > 2271634 n + 3986376 k n + 2 14 3 14 > 1263616 k n + 212048 k n + 4 14 5 14 > 43280 k n - 9472 k n - 6 14 15 15 > 1536 k n + 588212 n + 885712 k n + 2 15 3 15 > 189920 k n + 44960 k n + 4 15 16 > 13120 k n + 102480 n + 16 2 16 > 124864 k n + 5552 k n + 3 16 4 16 17 > 3456 k n + 1024 k n + 10752 n + 17 2 17 18 > 9536 k n - 2432 k n + 512 n + 18 2 18 > 256 k n - 256 k n ) S + k 2 3 > (4320 + 19712 k + 48456 k + 90528 k + 4 5 6 > 114160 k + 65992 k - 24396 k - 7 8 9 > 66980 k - 47708 k - 17396 k - 10 11 12 13 > 2892 k + 276 k + 220 k + 28 k + 2 > 64720 n + 279872 k n + 651788 k n + 3 4 > 1173116 k n + 1459540 k n + 5 6 7 > 887268 k n - 159130 k n - 646414 k n - 8 9 10 > 456882 k n - 160622 k n - 27354 k n + 11 12 13 > 362 k n + 1014 k n + 114 k n + 2 2 > 433896 n + 1752288 k n + 2 2 3 2 > 3852664 k n + 6778306 k n + 4 2 5 2 > 8446806 k n + 5424120 k n - 6 2 7 2 > 87794 k n - 2679130 k n - 8 2 9 2 > 1913246 k n - 645706 k n - 10 2 11 2 > 106406 k n - 2192 k n + 12 2 13 2 3 > 1920 k n + 154 k n + 1716500 n + 3 2 3 > 6323908 k n + 13023838 k n + 3 3 4 3 > 22914574 k n + 29205076 k n + 5 3 6 3 > 19852984 k n + 2301876 k n - 7 3 8 3 > 6184066 k n - 4617574 k n - 9 3 10 3 > 1500992 k n - 232242 k n - 11 3 12 3 13 3 > 6668 k n + 2350 k n + 84 k n + 4 4 > 4430744 n + 14247310 k n + 2 4 3 4 > 26988570 k n + 49683700 k n + 4 4 5 4 > 66954120 k n + 48272082 k n + 6 4 7 4 > 10511520 k n - 8486110 k n - 8 4 9 4 > 7101860 k n - 2247280 k n - 10 4 11 4 > 324942 k n - 9382 k n + 12 4 13 4 5 > 2184 k n + 16 k n + 7719622 n + 5 2 5 > 19451294 k n + 32010926 k n + 3 5 4 5 > 70106516 k n + 106563260 k n + 5 5 6 5 > 81917450 k n + 23996152 k n - 7 5 8 5 > 6448738 k n - 7231984 k n - 9 5 10 5 > 2265070 k n - 306892 k n - 11 5 12 5 6 > 9008 k n + 1360 k n + 8836204 n + 6 2 6 > 10914080 k n + 8736556 k n + 3 6 4 6 > 58757652 k n + 119320776 k n + 5 6 6 6 > 99309716 k n + 34427502 k n - 7 6 8 6 > 1171910 k n - 4876518 k n - 9 6 10 6 > 1554258 k n - 195528 k n - 11 6 12 6 7 > 5824 k n + 464 k n + 5418256 n - 7 2 7 > 15002790 k n - 42631338 k n + 3 7 4 7 > 12500846 k n + 92192214 k n + 5 7 6 7 > 86517542 k n + 33312146 k n + 7 7 8 7 > 2833296 k n - 2083460 k n - 9 7 10 7 > 720372 k n - 81488 k n - 11 7 12 7 8 > 2128 k n + 64 k n - 1331038 n - 8 2 8 > 45128232 k n - 91699556 k n - 3 8 4 8 > 37764846 k n + 45011186 k n + 5 8 6 8 > 53526158 k n + 22254424 k n + 7 8 8 8 > 3323928 k n - 478216 k n - 9 8 10 8 > 221104 k n - 21216 k n - 11 8 9 9 > 320 k n - 7123844 n - 59724272 k n - 2 9 3 9 > 105564150 k n - 58492696 k n + 4 9 5 9 > 8678604 k n + 22560238 k n + 6 9 7 9 > 10216428 k n + 1899192 k n - 8 9 9 9 10 9 > 4800 k n - 43488 k n - 3264 k n - 10 10 > 8755202 n - 52078018 k n - 2 10 3 10 > 81966408 k n - 46961106 k n - 4 10 5 10 > 5090040 k n + 5709242 k n + 6 10 7 10 > 3103688 k n + 656928 k n + 8 10 9 10 > 30208 k n - 5056 k n - 10 10 11 > 256 k n - 6722504 n - 11 2 11 > 32511702 k n - 45366632 k n - 3 11 4 11 > 24463844 k n - 4922086 k n + 5 11 6 11 > 392356 k n + 558656 k n + 7 11 8 11 > 139008 k n + 8128 k n - 9 11 12 > 256 k n - 3644458 n - 12 2 12 > 14858642 k n - 18049690 k n - 3 12 4 12 > 8516754 k n - 1966384 k n - 5 12 6 12 > 263408 k n + 37280 k n + 7 12 8 12 13 > 16320 k n + 768 k n - 1433162 n - 13 2 13 > 4949394 k n - 5064512 k n - 3 13 4 13 > 1887144 k n - 430736 k n - 5 13 6 13 7 13 > 97312 k n - 4672 k n + 768 k n - 14 14 > 404114 n - 1170658 k n - 2 14 3 14 > 953120 k n - 218464 k n - 4 14 5 14 > 45264 k n - 14144 k n - 6 14 15 15 > 768 k n - 77988 n - 186084 k n - 2 15 3 15 4 15 > 108016 k n + 592 k n - 256 k n - 5 15 16 16 > 768 k n - 9264 n - 17776 k n - 2 16 3 16 4 16 > 5568 k n + 3200 k n + 256 k n - 17 17 3 17 > 512 n - 768 k n + 256 k n ), 2 3 4 5 > (-32 k - 268 k - 570 k - 670 k - 6 7 8 9 > 736 k - 648 k - 300 k - 48 k + 10 11 > 6 k + 2 k - 88 n - 348 k n - 2 3 4 > 1002 k n - 3419 k n - 6156 k n - 5 6 7 > 6102 k n - 4485 k n - 2867 k n - 8 9 10 > 1148 k n - 158 k n + 29 k n + 11 2 2 > 8 k n - 1092 n - 3372 k n - 2 2 3 2 4 2 > 6911 k n - 15901 k n - 24273 k n - 5 2 6 2 7 2 > 21145 k n - 10960 k n - 4218 k n - 8 2 9 2 10 2 > 1304 k n - 156 k n + 42 k n + 11 2 3 3 > 8 k n - 5766 n - 13313 k n - 2 3 3 3 4 3 > 19901 k n - 35093 k n - 46966 k n - 5 3 6 3 7 3 > 37503 k n - 14598 k n - 1927 k n - 8 3 9 3 10 3 > 115 k n - 48 k n + 20 k n - 4 4 2 4 > 16843 n - 26669 k n - 24479 k n - 3 4 4 4 5 4 > 35914 k n - 47607 k n - 38221 k n - 6 4 7 4 8 4 > 12718 k n + 1073 k n + 886 k n + 9 4 10 4 5 > 24 k n + 16 k n - 29527 n - 5 2 5 3 5 > 23733 k n + 2249 k n - 5027 k n - 4 5 5 5 6 5 > 22067 k n - 23251 k n - 8693 k n + 7 5 8 5 9 5 > 1518 k n + 796 k n - 16 k n - 6 6 2 6 > 31224 n + 9927 k n + 48898 k n + 3 6 4 6 5 6 > 26132 k n - 70 k n - 8549 k n - 6 6 7 6 8 6 > 5378 k n + 484 k n + 368 k n - 7 7 2 7 > 18210 n + 52289 k n + 73928 k n + 3 7 4 7 5 7 > 29051 k n + 3313 k n - 1906 k n - 6 7 7 7 8 7 > 2620 k n - 112 k n + 64 k n - 8 8 2 8 > 3203 n + 65003 k n + 60312 k n + 3 8 4 8 5 8 > 16645 k n + 446 k n - 220 k n - 6 8 7 8 9 > 704 k n - 64 k n + 2677 n + 9 2 9 3 9 > 43469 k n + 29888 k n + 7490 k n - 4 9 6 9 10 > 60 k n - 64 k n + 1790 n + 10 2 10 3 10 > 16105 k n + 8016 k n + 2948 k n + 4 10 11 11 > 176 k n + 314 n + 2538 k n + 2 11 3 11 4 11 > 344 k n + 720 k n + 64 k n - 12 12 2 12 > 44 n - 172 k n - 368 k n + 3 12 13 13 2 13 > 64 k n - 16 n - 80 k n - 64 k n ) 3 2 3 4 > S + (6960 k + 38132 k + 78810 k + n 5 6 7 > 109992 k + 137326 k + 124090 k + 8 9 10 11 > 63578 k + 15806 k + 1314 k - 22 k + 12 13 > 12 k + 2 k + 14280 n + 51436 k n + 2 3 4 > 170666 k n + 529055 k n + 938285 k n + 5 6 > 1077456 k n + 1020902 k n + 7 8 9 > 774714 k n + 366868 k n + 86364 k n + 10 11 12 > 6506 k n - 81 k n + 53 k n + 13 2 2 > 8 k n + 184580 n + 534724 k n + 2 2 3 2 > 1181495 k n + 2702571 k n + 4 2 5 2 > 4206690 k n + 4251665 k n + 6 2 7 2 > 3230328 k n + 2015921 k n + 8 2 9 2 > 870344 k n + 192597 k n + 10 2 11 2 12 2 > 12109 k n - 114 k n + 70 k n + 13 2 3 3 > 8 k n + 1034766 n + 2329333 k n + 2 3 3 3 > 3743857 k n + 6873574 k n + 4 3 5 3 > 9611528 k n + 8940421 k n + 6 3 7 3 > 5703088 k n + 2868018 k n + 8 3 9 3 > 1128746 k n + 239424 k n + 10 3 11 3 12 3 > 11053 k n - 74 k n + 28 k n + 4 4 > 3318291 n + 5447753 k n + 2 4 3 4 > 5837232 k n + 9077761 k n + 4 4 5 4 > 12066575 k n + 11025310 k n + 6 4 7 4 > 6171310 k n + 2363251 k n + 8 4 9 4 > 857288 k n + 187675 k n + 10 4 11 4 12 4 > 5080 k n + 12 k n + 16 k n + 5 5 > 6754918 n + 6843686 k n + 2 5 3 5 > 2526245 k n + 4279484 k n + 4 5 5 5 > 7337212 k n + 8038974 k n + 6 5 7 5 > 4294779 k n + 1030062 k n + 8 5 9 5 10 5 > 346798 k n + 97282 k n + 992 k n - 11 5 6 6 > 16 k n + 9139460 n + 2363828 k n - 2 6 3 6 > 7218146 k n - 4907057 k n - 4 6 5 6 > 476956 k n + 3045907 k n + 6 6 7 6 > 2065810 k n + 107252 k n + 8 6 9 6 10 6 > 22752 k n + 32580 k n + 64 k n + 7 7 > 8306725 n - 7025819 k n - 2 7 3 7 > 16541944 k n - 10137032 k n - 4 7 5 7 > 4201663 k n + 88406 k n + 6 7 7 7 > 828868 k n - 94324 k n - 8 7 9 7 8 > 45568 k n + 6272 k n + 4896137 n - 8 2 8 > 14386964 k n - 18383746 k n - 3 8 4 8 > 8652165 k n - 3072362 k n - 5 8 6 8 > 445032 k n + 342696 k n - 7 8 8 8 9 8 > 32688 k n - 21072 k n + 512 k n + 9 9 > 1571120 n - 14534436 k n - 2 9 3 9 > 13293101 k n - 4738689 k n - 4 9 5 9 > 1016805 k n - 183268 k n + 6 9 7 9 8 9 > 127736 k n + 1472 k n - 3840 k n - 10 10 > 39594 n - 9336352 k n - 2 10 3 10 > 6642091 k n - 2014378 k n - 4 10 5 10 > 132818 k n - 24808 k n + 6 10 7 10 8 10 > 32208 k n + 2112 k n - 256 k n - 11 11 > 289411 n - 3943826 k n - 2 11 3 11 > 2225305 k n - 745954 k n - 4 11 5 11 6 11 > 1564 k n + 1248 k n + 4480 k n + 7 11 12 12 > 256 k n - 130558 n - 1038989 k n - 2 12 3 12 > 420208 k n - 225884 k n - 4 12 5 12 6 12 > 4800 k n + 448 k n + 256 k n - 13 13 2 13 > 25174 n - 136070 k n - 4856 k n - 3 13 4 13 14 > 46768 k n - 2304 k n - 612 n + 14 2 14 3 14 > 3484 k n + 17584 k n - 5440 k n - 4 14 15 15 > 256 k n + 544 n + 3488 k n + 2 15 3 15 16 > 3712 k n - 256 k n + 64 n + 16 2 16 2 > 320 k n + 256 k n ) S + n 2 3 4 > (12960 k - 41568 k + 4776 k + 131512 k + 5 6 7 > 278904 k + 39046 k - 205322 k - 8 9 10 > 149780 k - 48152 k - 15900 k - 11 12 13 > 5552 k - 894 k - 30 k + 2 3 > 128880 k n - 360112 k n + 454188 k n + 4 5 > 1575156 k n + 2832336 k n + 6 7 > 556165 k n - 1920635 k n - 8 9 > 1469039 k n - 444161 k n - 10 11 12 > 112177 k n - 34239 k n - 5157 k n - 13 2 2 > 149 k n + 12960 n + 564744 k n - 2 2 3 2 > 1081936 k n + 4344948 k n + 4 2 5 2 > 8869684 k n + 13188518 k n + 6 2 7 2 > 3332970 k n - 7713076 k n - 8 2 9 2 > 6124679 k n - 1719623 k n - 10 2 11 2 > 315572 k n - 80724 k n - 12 2 13 2 3 > 11021 k n - 245 k n + 94992 n + 3 2 3 > 1646948 k n - 167880 k n + 3 3 4 3 > 18622112 k n + 29887892 k n + 5 3 6 3 > 37492811 k n + 12187568 k n - 7 3 8 3 > 16943687 k n - 14338229 k n - 9 3 10 3 > 3774117 k n - 481520 k n - 11 3 12 3 > 97029 k n - 11103 k n - 13 3 4 4 > 152 k n + 15224 n + 4047004 k n + 2 4 3 4 > 7577070 k n + 45243804 k n + 4 4 5 4 > 66048398 k n + 72441137 k n + 6 4 7 4 > 29952285 k n - 21411597 k n - 8 4 9 4 > 20970504 k n - 5310937 k n - 10 4 11 4 > 464259 k n - 71349 k n - 12 4 13 4 5 > 5862 k n - 32 k n - 2167028 n + 5 2 5 > 7935992 k n + 21553011 k n + 3 5 4 5 > 65782739 k n + 100996163 k n + 5 5 6 5 > 99840049 k n + 50369615 k n - 7 5 8 5 > 14093098 k n - 20139808 k n - 9 5 10 5 > 5065714 k n - 292239 k n - 11 5 12 5 > 35900 k n - 1620 k n - 6 6 > 10752976 n + 8997744 k n + 2 6 3 6 > 22566426 k n + 49295848 k n + 4 6 5 6 > 110239209 k n + 100510333 k n + 6 6 7 6 > 58717298 k n - 732478 k n - 8 6 9 6 > 12934908 k n - 3381956 k n - 10 6 11 6 > 114178 k n - 12440 k n - 12 6 7 7 > 192 k n - 27317748 n - 1326443 k n - 2 7 3 7 > 12742444 k n - 10071169 k n + 4 7 5 7 > 85850591 k n + 74555624 k n + 6 7 7 7 > 48083096 k n + 7301480 k n - 8 7 9 7 > 5461734 k n - 1608188 k n - 10 7 11 7 > 24676 k n - 2640 k n - 8 8 > 43395782 n - 23265773 k n - 2 8 3 8 > 72651393 k n - 74576235 k n + 4 8 5 8 > 44332190 k n + 40346382 k n + 6 8 7 8 > 27847804 k n + 7086314 k n - 8 8 9 8 > 1379396 k n - 540864 k n - 10 8 11 8 9 > 2240 k n - 256 k n - 45418712 n - 9 2 9 > 41459448 k n - 110100518 k n - 3 9 4 9 > 98819735 k n + 9814507 k n + 5 9 6 9 > 15165605 k n + 11345114 k n + 7 9 8 9 > 3618128 k n - 125752 k n - 9 9 10 > 122624 k n - 30706450 n - 10 2 10 > 41433230 k n - 97450334 k n - 3 10 4 10 > 76825974 k n - 5872591 k n + 5 10 6 10 > 3295647 k n + 3175316 k n + 7 10 8 10 > 1135856 k n + 33728 k n - 9 10 11 > 16704 k n - 11297570 n - 11 2 11 > 26836563 k n - 55825552 k n - 3 11 4 11 > 39008819 k n - 7062859 k n - 5 11 6 11 > 36920 k n + 579176 k n + 7 11 8 11 > 218144 k n + 11328 k n - 9 11 12 > 1024 k n + 519832 n - 12 2 12 > 11583919 k n - 20604505 k n - 3 12 4 12 > 12693605 k n - 3607270 k n - 5 12 6 12 > 295520 k n + 61632 k n + 7 12 8 12 > 23232 k n + 1024 k n + 13 13 > 3226338 n - 3247630 k n - 2 13 3 13 > 4397787 k n - 2257940 k n - 4 13 5 13 > 1108596 k n - 100160 k n + 6 13 7 13 14 > 2880 k n + 1024 k n + 1966328 n - 14 2 14 > 534326 k n - 275122 k n - 3 14 4 14 > 27224 k n - 209408 k n - 5 14 15 15 > 15808 k n + 657056 n - 34100 k n + 2 15 3 15 > 104572 k n + 79536 k n - 4 15 5 15 16 > 22336 k n - 1024 k n + 132656 n + 16 2 16 > 2880 k n + 26624 k n + 3 16 4 16 17 > 15680 k n - 1024 k n + 15232 n + 17 2 17 3 17 > 448 k n + 1984 k n + 1024 k n + 18 > 768 n ) S S + n k 2 3 4 5 > (12960 k - 15648 k - 600 k - 1544 k + 6 7 8 > 9144 k + 118566 k + 102088 k + 9 10 11 > 31206 k + 13794 k + 5642 k + 12 13 14 2 > 852 k + 18 k + 2 k + 219600 k n - 3 4 5 > 373360 k n - 422884 k n - 38892 k n + 6 7 > 348000 k n + 1711444 k n + 8 9 > 1384868 k n + 395864 k n + 10 11 12 > 121640 k n + 35596 k n + 4336 k n + 13 14 2 > 84 k n + 8 k n - 12960 n - 2 2 2 > 96000 k n + 1254736 k n - 3 2 4 2 > 3316328 k n - 4929000 k n - 5 2 6 2 > 324928 k n + 2757964 k n + 7 2 8 2 > 10203584 k n + 7751384 k n + 9 2 10 2 > 2002410 k n + 475810 k n + 11 2 12 2 13 2 > 95036 k n + 8664 k n + 170 k n + 14 2 3 3 > 10 k n - 107952 n - 751120 k n + 2 3 3 3 > 3341176 k n - 16020720 k n - 4 3 5 3 > 27087508 k n - 2610398 k n + 6 3 7 3 > 9101078 k n + 33854952 k n + 8 3 9 3 > 24491168 k n + 5559318 k n + 10 3 11 3 > 1087242 k n + 140244 k n + 12 3 13 3 14 3 > 9280 k n + 228 k n + 4 k n - 4 4 > 129656 n - 1664816 k n + 2 4 3 4 > 4305976 k n - 48204154 k n - 4 4 5 4 > 92313490 k n - 14216048 k n + 6 4 7 4 > 11461700 k n + 70019270 k n + 8 4 9 4 > 49681746 k n + 9666386 k n + 10 4 11 4 > 1623686 k n + 119294 k n + 12 4 13 4 5 > 6184 k n + 204 k n + 2064396 n + 5 2 5 > 3468188 k n + 3329224 k n - 3 5 4 5 > 95045736 k n - 215895916 k n - 5 5 6 5 > 45860770 k n - 12217766 k n + 7 5 8 5 > 94582732 k n + 69152888 k n + 9 5 10 5 > 11230358 k n + 1692910 k n + 11 5 12 5 13 5 > 52016 k n + 2964 k n + 96 k n + 6 6 > 13205304 n + 30852708 k n + 2 6 3 6 > 11027536 k n - 121511668 k n - 4 6 5 6 > 363315430 k n - 91479668 k n - 6 6 7 6 > 70258328 k n + 83497948 k n + 8 6 9 6 > 68529526 k n + 8932274 k n + 10 6 11 6 > 1268814 k n + 3292 k n + 12 6 13 6 7 > 1216 k n + 16 k n + 40968092 n + 7 2 7 > 95504706 k n + 45029938 k n - 3 7 4 7 > 86293524 k n - 451458228 k n - 5 7 6 7 > 117917244 k n - 128597904 k n + 7 7 8 7 > 44662144 k n + 49391508 k n + 9 7 10 7 > 4833528 k n + 674360 k n - 11 7 12 7 8 > 7328 k n + 400 k n + 78865886 n + 8 2 8 > 188885550 k n + 104298438 k n + 3 8 4 8 > 7077752 k n - 420360520 k n - 5 8 6 8 > 99207462 k n - 141443530 k n + 7 8 8 8 > 8883928 k n + 26206512 k n + 9 8 10 8 > 1723480 k n + 241552 k n - 11 8 12 8 9 > 3184 k n + 64 k n + 96987408 n + 9 2 9 > 272387014 k n + 157589990 k n + 3 9 4 9 > 103236768 k n - 294669028 k n - 5 9 6 9 > 51367448 k n - 105139208 k n - 7 9 8 9 > 6204736 k n + 10265192 k n + 9 9 10 9 > 380624 k n + 53520 k n - 11 9 10 > 448 k n + 65306578 n + 10 2 10 > 300981442 k n + 169060350 k n + 3 10 4 10 > 147711800 k n - 153965140 k n - 5 10 6 10 > 11191094 k n - 54890634 k n - 7 10 8 10 > 6231096 k n + 2926528 k n + 9 10 10 10 > 46240 k n + 6272 k n - 11 11 > 7133652 n + 258472142 k n + 2 11 3 11 > 132043814 k n + 131512380 k n - 4 11 5 11 > 57697780 k n + 4899092 k n - 6 11 7 11 > 20236508 k n - 2706432 k n + 8 11 9 11 > 579360 k n + 2304 k n + 10 11 12 > 256 k n - 72569222 n + 12 2 12 > 171445106 k n + 73256434 k n + 3 12 4 12 > 84243626 k n - 13799448 k n + 5 12 6 12 > 5280588 k n - 5160544 k n - 7 12 8 12 > 683616 k n + 71296 k n - 13 13 > 92230180 n + 86129310 k n + 2 13 3 13 > 25942738 k n + 39902832 k n - 4 13 5 13 > 1124044 k n + 2219328 k n - 6 13 7 13 > 862560 k n - 97664 k n + 8 13 14 > 4096 k n - 70316714 n + 14 2 14 > 31723866 k n + 3330642 k n + 3 14 4 14 > 13802908 k n + 502592 k n + 5 14 6 14 > 530384 k n - 84224 k n - 7 14 15 > 6144 k n - 36911520 n + 15 2 15 > 8120496 k n - 1890336 k n + 3 15 4 15 > 3360544 k n + 204880 k n + 5 15 6 15 > 70656 k n - 3584 k n - 16 16 > 13725536 n + 1300816 k n - 2 16 3 16 > 1233072 k n + 539344 k n + 4 16 5 16 > 32576 k n + 4096 k n - 17 17 > 3571824 n + 95504 k n - 2 17 3 17 > 336816 k n + 50496 k n + 4 17 18 18 > 2048 k n - 620608 n - 3200 k n - 2 18 3 18 19 > 47488 k n + 2048 k n - 64768 n - 19 2 19 20 2 > 768 k n - 2816 k n - 3072 n ) S + k 2 3 > (-12960 - 10272 k - 33624 k - 345088 k - 4 5 6 > 1107752 k - 1976030 k - 2171034 k - 7 8 9 > 1419670 k - 475486 k - 25176 k + 10 11 12 13 > 33112 k + 9178 k + 698 k + 18 k + 14 > 6 k - 320400 n - 600944 k n - 2 3 > 1584172 k n - 5846820 k n - 4 5 > 13677178 k n - 20781184 k n - 6 7 > 20554425 k n - 12574810 k n - 8 9 > 4202797 k n - 432224 k n + 10 11 12 > 146873 k n + 42434 k n + 2299 k n + 13 14 2 > 28 k n + 24 k n - 3022536 n - 2 2 2 > 5589192 k n - 12301460 k n - 3 2 4 2 > 34858790 k n - 69414688 k n - 5 2 6 2 > 95587048 k n - 87226341 k n - 7 2 8 2 > 49940272 k n - 16290387 k n - 9 2 10 2 > 2112879 k n + 244251 k n + 11 2 12 2 13 2 > 81090 k n + 1821 k n - 61 k n + 14 2 3 3 > 28 k n - 15716636 n - 24174284 k n - 2 3 3 3 > 41623950 k n - 103614614 k n - 4 3 5 3 > 189693641 k n - 250006922 k n - 6 3 7 3 > 216811666 k n - 117056997 k n - 8 3 9 3 > 36829059 k n - 5204110 k n + 10 3 11 3 > 185976 k n + 95925 k n - 12 3 13 3 14 3 > 588 k n - 114 k n + 8 k n - 4 4 > 52067812 n - 59162468 k n - 2 4 3 4 > 66504892 k n - 161375806 k n - 4 4 5 4 > 297328093 k n - 405105531 k n - 6 4 7 4 > 346936611 k n - 178779313 k n - 8 4 9 4 > 54136537 k n - 7909033 k n - 10 4 11 4 > 12777 k n + 78211 k n - 12 4 13 4 5 > 1846 k n - 36 k n - 118490176 n - 5 2 5 > 81119346 k n - 9432207 k n - 3 5 4 5 > 81095288 k n - 233940297 k n - 5 5 6 5 > 410175503 k n - 371587762 k n - 7 5 8 5 > 185834020 k n - 54189818 k n - 9 5 10 5 > 7983291 k n - 175770 k n + 11 5 12 5 > 43294 k n - 1380 k n - 6 6 > 193879954 n - 33470324 k n + 2 6 3 6 > 184953603 k n + 185514512 k n + 4 6 5 6 > 33103501 k n - 230560977 k n - 6 6 7 6 > 267354016 k n - 133684536 k n - 8 6 9 6 > 37567400 k n - 5465568 k n - 10 6 11 6 > 179154 k n + 15516 k n - 12 6 7 > 480 k n - 235309007 n + 7 2 7 > 100873844 k n + 436877830 k n + 3 7 4 7 > 479469969 k n + 309965536 k n - 5 7 6 7 > 12434310 k n - 123168206 k n - 7 7 8 7 > 66303728 k n - 18024624 k n - 9 7 10 7 > 2529644 k n - 92300 k n + 11 7 12 7 8 > 3296 k n - 64 k n - 216824155 n + 8 2 8 > 249556323 k n + 573476275 k n + 3 8 4 8 > 582592659 k n + 391994768 k n + 5 8 6 8 > 95567528 k n - 28723150 k n - 7 8 8 8 > 21985558 k n - 5887780 k n - 9 8 10 8 > 774632 k n - 26688 k n + 11 8 9 > 320 k n - 154557390 n + 9 2 9 > 311048163 k n + 510186981 k n + 3 9 4 9 > 459500110 k n + 285192921 k n + 5 9 6 9 > 87951372 k n + 3339296 k n - 7 9 8 9 > 4445044 k n - 1261224 k n - 9 9 10 9 > 149552 k n - 4096 k n - 10 10 > 86590386 n + 259168580 k n + 2 10 3 10 > 325919289 k n + 254130702 k n + 4 10 5 10 > 135680643 k n + 43158543 k n + 6 10 7 10 > 5244256 k n - 372232 k n - 8 10 9 10 > 164416 k n - 16384 k n - 10 10 11 > 256 k n - 38626697 n + 11 2 11 > 154442056 k n + 151922542 k n + 3 11 4 11 > 101196963 k n + 42460632 k n + 5 11 6 11 > 13150358 k n + 1966304 k n + 7 11 8 11 > 45472 k n - 11136 k n - 9 11 12 > 768 k n - 13833221 n + 12 2 12 > 66851991 k n + 51113897 k n + 3 12 4 12 > 29281059 k n + 8072762 k n + 5 12 6 12 > 2441772 k n + 396352 k n + 7 12 8 12 13 > 14016 k n - 256 k n - 3975858 n + 13 2 13 > 20736427 k n + 11835826 k n + 3 13 4 13 > 6197750 k n + 642524 k n + 5 13 6 13 > 235392 k n + 43520 k n + 7 13 14 > 1024 k n - 904560 n + 14 2 14 > 4396670 k n + 1628182 k n + 3 14 4 14 > 976940 k n - 67488 k n + 5 14 6 14 15 > 4160 k n + 2048 k n - 157740 n + 15 2 15 > 564648 k n + 46116 k n + 3 15 4 15 > 117888 k n - 19520 k n - 5 15 16 16 > 768 k n - 19872 n + 26912 k n - 2 16 3 16 > 26624 k n + 10496 k n - 4 16 17 17 > 1280 k n - 1616 n - 2640 k n - 2 17 3 17 18 > 4608 k n + 512 k n - 64 n - 18 2 18 > 320 k n - 256 k n ) S + n 2 3 > (12960 k + 10272 k - 135528 k - 4 5 6 > 16224 k + 84504 k + 422814 k + 7 8 9 > 389586 k + 50068 k - 60060 k - 10 11 12 > 39758 k - 20180 k - 6380 k - 13 14 15 > 848 k - 24 k - 2 k - 12960 n + 2 3 > 162480 k n + 101112 k n - 2222788 k n + 4 5 > 32436 k n + 1753048 k n + 6 7 > 5739394 k n + 4894918 k n + 8 9 > 423878 k n - 842530 k n - 10 11 > 395182 k n - 128030 k n - 12 13 14 > 32498 k n - 4106 k n - 108 k n - 15 2 2 > 8 k n - 172752 n + 781592 k n - 2 2 3 2 > 31972 k n - 16785664 k n + 4 2 5 2 > 283652 k n + 12972896 k n + 6 2 7 2 > 34698864 k n + 28325946 k n + 8 2 9 2 > 2591484 k n - 4399508 k n - 10 2 11 2 > 1761304 k n - 371020 k n - 12 2 13 2 14 2 > 62456 k n - 7596 k n - 200 k n - 15 2 3 3 > 10 k n - 747176 n + 1547940 k n - 2 3 3 3 > 5723384 k n - 78954088 k n - 4 3 5 3 > 5963750 k n + 50520088 k n + 6 3 7 3 > 124511874 k n + 99126256 k n + 8 3 9 3 > 11627708 k n - 12249546 k n - 10 3 11 3 > 4609038 k n - 693724 k n - 12 3 13 3 14 3 > 54450 k n - 7374 k n - 240 k n - 15 3 4 4 > 4 k n + 723044 n + 140472 k n - 2 4 3 4 > 41508786 k n - 261161258 k n - 4 4 5 4 > 58674770 k n + 115870700 k n + 6 4 7 4 > 296178944 k n + 232872632 k n + 8 4 9 4 > 34263198 k n - 20943180 k n - 10 4 11 4 > 7837508 k n - 955766 k n - 12 4 13 4 14 4 > 9172 k n - 4560 k n - 204 k n + 5 5 > 22251636 n + 73104 k n - 2 5 3 5 > 157033182 k n - 642370014 k n - 4 5 5 5 > 251775962 k n + 150169832 k n + 6 5 7 5 > 490731500 k n + 386866710 k n + 8 5 9 5 > 67020200 k n - 23373526 k n - 10 5 11 5 > 9089080 k n - 1006006 k n + 12 5 13 5 14 5 > 27088 k n - 2292 k n - 96 k n + 6 6 > 117930256 n + 36231672 k n - 2 6 3 6 > 373468160 k n - 1201492050 k n - 4 6 5 6 > 659603114 k n + 54981792 k n + 6 6 7 6 > 575435906 k n + 467018976 k n + 8 6 9 6 > 90219578 k n - 17224760 k n - 10 6 11 6 > 7311342 k n - 794322 k n + 12 6 13 6 14 6 > 29348 k n - 1104 k n - 16 k n + 7 7 > 368824642 n + 183177680 k n - 2 7 3 7 > 587735670 k n - 1716272704 k n - 4 7 5 7 > 1173825856 k n - 179599752 k n + 6 7 7 7 > 470799660 k n + 414801960 k n + 8 7 9 7 > 85806890 k n - 8044534 k n - 10 7 11 7 > 4047660 k n - 448936 k n + 12 7 13 7 > 14640 k n - 400 k n + 8 8 > 801171326 n + 484881560 k n - 2 8 3 8 > 598627134 k n - 1860233832 k n - 4 8 5 8 > 1488018344 k n - 406331360 k n + 6 8 7 8 > 250422868 k n + 271612810 k n + 8 8 9 8 > 58426628 k n - 2021692 k n - 10 8 11 8 > 1496968 k n - 171072 k n + 12 8 13 8 9 > 3888 k n - 64 k n + 1287049784 n + 9 2 9 > 849083818 k n - 322676266 k n - 3 9 4 9 > 1504101720 k n - 1371888890 k n - 5 9 6 9 > 462211574 k n + 62006708 k n + 7 9 8 9 > 130179296 k n + 28557568 k n - 9 9 10 9 > 41784 k n - 349344 k n - 11 9 12 9 > 40656 k n + 448 k n + 10 10 > 1578234906 n + 1068576360 k n + 2 10 3 10 > 65358678 k n - 874936844 k n - 4 10 5 10 > 922017832 k n - 343845468 k n - 6 10 7 10 > 21314032 k n + 44847490 k n + 8 10 9 10 > 9956456 k n + 128224 k n - 10 10 11 10 > 45984 k n - 5248 k n + 11 11 > 1502760832 n + 1003598674 k n + 2 11 3 11 > 303766042 k n - 334745532 k n - 4 11 5 11 > 443524678 k n - 177068966 k n - 6 11 7 11 > 28567384 k n + 10711932 k n + 8 11 9 11 > 2434400 k n + 33888 k n - 10 11 11 11 > 2560 k n - 256 k n + 12 12 > 1120341910 n + 715493556 k n + 2 12 3 12 > 307663432 k n - 58465474 k n - 4 12 5 12 > 144249192 k n - 63131848 k n - 6 12 7 12 > 14109772 k n + 1644800 k n + 8 12 9 12 > 402528 k n + 2944 k n + 13 13 > 655015244 n + 388885446 k n + 2 13 3 13 > 188260256 k n + 15144158 k n - 4 13 5 13 > 26257824 k n - 14889268 k n - 6 13 7 13 > 4184608 k n + 132448 k n + 8 13 14 > 41344 k n + 298952998 n + 14 2 14 > 160181744 k n + 78057958 k n + 3 14 4 14 > 12776890 k n + 248356 k n - 5 14 6 14 > 2013456 k n - 778832 k n + 7 14 8 14 > 768 k n + 2048 k n + 15 15 > 105298974 n + 49160810 k n + 2 15 3 15 > 22271572 k n + 3421680 k n + 4 15 5 15 > 1500272 k n - 68560 k n - 6 15 7 15 > 84480 k n - 512 k n + 16 16 > 28041976 n + 10889076 k n + 2 16 3 16 > 4246368 k n + 285888 k n + 4 16 5 16 > 385904 k n + 19136 k n - 6 16 17 > 4096 k n + 5455856 n + 17 2 17 > 1643936 k n + 499872 k n - 3 17 4 17 > 63120 k n + 44736 k n + 5 17 18 18 > 2048 k n + 731152 n + 151296 k n + 2 18 3 18 > 30208 k n - 17536 k n + 4 18 19 19 > 2048 k n + 60288 n + 6400 k n + 2 19 3 19 20 > 512 k n - 1280 k n + 2304 n ) S + k 2 3 > (25920 - 18336 k - 68688 k + 159224 k + 4 5 6 > 648824 k + 926076 k + 419890 k - 7 8 9 > 415128 k - 657368 k - 375818 k - 10 11 12 > 97452 k - 298 k + 5344 k + 13 14 15 > 758 k + 10 k + 2 k + 452160 n - 2 3 > 205424 k n - 656240 k n + 2903028 k n + 4 5 > 9104492 k n + 11974436 k n + 6 7 > 5393030 k n - 4319824 k n - 8 9 > 6680066 k n - 3579860 k n - 10 11 > 903050 k n - 35828 k n + 12 13 14 > 29638 k n + 3432 k n + 4 k n + 15 2 2 > 8 k n + 3430848 n - 1854744 k n - 2 2 3 2 > 3857508 k n + 21163820 k n + 4 2 5 2 > 58302420 k n + 72979072 k n + 6 2 7 2 > 34185306 k n - 18625682 k n - 8 2 9 2 > 30193604 k n - 15265210 k n - 10 2 11 2 > 3683688 k n - 217248 k n + 12 2 13 2 14 2 > 71992 k n + 5358 k n - 118 k n + 15 2 3 3 > 10 k n + 14829584 n - 14987844 k n - 2 3 3 3 > 22726532 k n + 83163128 k n + 4 3 5 3 > 223754052 k n + 275668092 k n + 6 3 7 3 > 137660380 k n - 40844126 k n - 8 3 9 3 > 79708892 k n - 38535340 k n - 10 3 11 3 > 8840548 k n - 592034 k n + 12 3 13 3 > 110960 k n + 3016 k n - 14 3 15 3 4 > 268 k n + 4 k n + 39390908 n - 4 2 4 > 84839336 k n - 115156382 k n + 3 4 4 4 > 189575592 k n + 568176138 k n + 5 4 6 4 > 712589872 k n + 380434030 k n - 7 4 8 4 > 38234650 k n - 135638674 k n - 9 4 10 4 > 63952206 k n - 13996116 k n - 11 4 12 4 > 986100 k n + 123716 k n - 13 4 14 4 5 > 932 k n - 236 k n + 60731672 n - 5 2 5 > 324695004 k n - 420676490 k n + 3 5 4 5 > 219613704 k n + 992388160 k n + 5 5 6 5 > 1317515112 k n + 745608280 k n + 7 5 8 5 > 28215586 k n - 154764108 k n - 9 5 10 5 > 73143112 k n - 15363852 k n - 11 5 12 5 > 1124290 k n + 99086 k n - 13 5 14 5 6 > 2368 k n - 96 k n + 25029912 n - 6 2 6 > 873436704 k n - 1086530394 k n - 3 6 4 6 > 52719094 k n + 1190699914 k n + 5 6 6 6 > 1777693816 k n + 1056316934 k n + 7 6 8 6 > 136806324 k n - 118634654 k n - 9 6 10 6 > 58777514 k n - 11871996 k n - 11 6 12 6 > 899014 k n + 54104 k n - 13 6 14 6 7 > 1520 k n - 16 k n - 125098416 n - 7 2 7 > 1718711796 k n - 2035651288 k n - 3 7 4 7 > 697981746 k n + 919753582 k n + 5 7 6 7 > 1759889464 k n + 1094476546 k n + 7 7 8 7 > 206049692 k n - 58140410 k n - 9 7 10 7 > 33277736 k n - 6420358 k n - 11 7 12 7 > 494748 k n + 19072 k n - 13 7 8 > 480 k n - 371255314 n - 8 2 8 > 2548790846 k n - 2840718550 k n - 3 8 4 8 > 1406022180 k n + 320164284 k n + 5 8 6 8 > 1266287902 k n + 832792922 k n + 7 8 8 8 > 188067638 k n - 14450644 k n - 9 8 10 8 > 13155380 k n - 2380204 k n - 11 8 12 8 13 8 > 180272 k n + 3984 k n - 64 k n - 9 9 > 593118178 n - 2907407140 k n - 2 9 3 9 > 3009897766 k n - 1708637582 k n - 4 9 5 9 > 193052364 k n + 639220296 k n + 6 9 7 9 > 462481424 k n + 116594368 k n + 8 9 9 9 > 1662084 k n - 3564848 k n - 10 9 11 9 > 583152 k n - 40896 k n + 12 9 10 > 384 k n - 660412786 n - 10 2 10 > 2584228742 k n - 2449644886 k n - 3 10 4 10 > 1450649476 k n - 362861676 k n + 5 10 6 10 > 202756702 k n + 183096710 k n + 7 10 8 10 > 50849078 k n + 2876416 k n - 9 10 10 10 > 645296 k n - 88672 k n - 11 10 11 > 5120 k n - 549846762 n - 11 2 11 > 1801693100 k n - 1538273966 k n - 3 11 4 11 > 899725322 k n - 270365528 k n + 5 11 6 11 > 21219556 k n + 48574080 k n + 7 11 8 11 > 15655052 k n + 1159072 k n - 9 11 10 11 > 76032 k n - 7424 k n - 11 11 12 > 256 k n - 351494738 n - 12 2 12 > 986030998 k n - 743340848 k n - 3 12 4 12 > 412477716 k n - 124598192 k n - 5 12 6 12 > 14169764 k n + 7070300 k n + 7 12 8 12 > 3330432 k n + 256416 k n - 9 12 10 12 > 5760 k n - 256 k n - 13 13 > 173847262 n - 421360344 k n - 2 13 3 13 > 273399128 k n - 138739910 k n - 4 13 5 13 > 37174386 k n - 8344512 k n - 6 13 7 13 > 89696 k n + 463360 k n + 8 13 9 13 > 32000 k n - 256 k n - 14 14 > 66205838 n - 138851794 k n - 2 14 3 14 > 74843028 k n - 33354394 k n - 4 14 5 14 > 6603624 k n - 2231536 k n - 6 14 7 14 > 254288 k n + 37376 k n + 8 14 15 > 1792 k n - 19090126 n - 15 2 15 > 34515908 k n - 14638222 k n - 3 15 4 15 > 5432564 k n - 414048 k n - 5 15 6 15 > 335968 k n - 45056 k n + 7 15 16 > 1280 k n - 4035136 n - 16 2 16 > 6240356 k n - 1891204 k n - 3 16 4 16 > 537424 k n + 84432 k n - 5 16 6 16 17 > 26432 k n - 2816 k n - 589952 n - 17 2 17 > 771280 k n - 133968 k n - 3 17 4 17 > 24384 k n + 19840 k n - 5 17 18 18 > 768 k n - 53296 n - 57968 k n - 2 18 4 18 19 > 1856 k n + 1280 k n - 2240 n - 19 2 19 > 1984 k n + 256 k n )} In[39]:= CreativeTelescoping[Binomial[n,k], S[k]-1, {S[n]}] Annihilator called with Gamma[1 +\ > n]/(Gamma[1 + k]*Gamma[1 - k + n]). Annihilator: The factors that contain\ > not-to-be-evaluated elements are {} Annihilator: The remaining factors are\ > {Gamma[1 + k]^(-1), Gamma[1 + n], Gamma[1 - k\ > + n]^(-1)} Annihilator: Factors that are not\ > hypergeometric and hyperexponential: {} CreativeTelescoping: Trying d = 0, ansatz =\ > eta[0]**1 + (-1 + S[k])**phi[1][k]**1 LocalOreReduce: Reducing {0, 0} LocalOreReduce: Reducing {1, 0} Start to solve scalar equation... RSolveRational: got a recurrence of order\ > 1 RSolvePolynomial: degree bound = 0 Solved scalar equation. CreativeTelescoping: Trying d = 1, ansatz =\ > eta[0]**1 + eta[1]**S[n] + (-1 +\ > S[k])**phi[1][k]**1 LocalOreReduce: Reducing {0, 1} Start to solve scalar equation... RSolveRational: got a recurrence of order\ > 1 RSolvePolynomial: degree bound = 1 Solved scalar equation. k Out[39]= {{S - 2}, {-----------}} n -1 + k - n In[40]:= Printlevel = 0 Out[40]= 0 In[41]:= CreativeTelescoping[Binomial[n,k], S[k]-1, {S[n]}] k Out[41]= {{S - 2}, {-----------}} n -1 + k - n In[42]:= CreativeTelescoping[Binomial[n,k]^2, S[k]-1, {S[n]}] Out[42]= {{(1 + n) S + (-2 - 4 n)}, n 2 3 2 3 k - 2 k + 3 k n > {--------------------}} 2 (-1 + k - n) In[43]:= CreativeTelescoping[Binomial[n,k]^4, S[k]-1, {S[n]}] 2 3 2 Out[43]= {{(8 + 12 n + 6 n + n ) S + n 2 3 > (-42 - 82 n - 54 n - 12 n ) S + n 2 3 > (-60 - 188 n - 192 n - 64 n )}, 4 5 6 > {((1 + n) (1080 k - 2256 k + 1980 k - 7 8 9 4 > 900 k + 210 k - 20 k + 4300 k n - 5 6 7 > 7520 k n + 5322 k n - 1844 k n + 8 9 4 2 > 298 k n - 16 k n + 7030 k n - 5 2 6 2 > 9892 k n + 5298 k n - 7 2 8 2 4 3 > 1244 k n + 104 k n + 6045 k n - 5 3 6 3 7 3 > 6420 k n + 2314 k n - 276 k n + 4 4 5 4 6 4 > 2885 k n - 2056 k n + 374 k n + 4 5 5 5 4 6 > 725 k n - 260 k n + 75 k n )) / 2 2 4 > (2 - 3 k + k + 3 n - 2 k n + n ) }} In[44]:= First[CreativeTelescoping[Binomial[n,k]^4, S[k]-1, {S[n]}]] 2 3 2 Out[44]= {(8 + 12 n + 6 n + n ) S + n 2 3 > (-42 - 82 n - 54 n - 12 n ) S + n 2 3 > (-60 - 188 n - 192 n - 64 n )} In[45]:= First[CreativeTelescoping[Binomial[n,k]^2 Binomial[n+k,k]^2, S[k]-1, {S[n]}]] 2 3 2 Out[45]= {(8 + 12 n + 6 n + n ) S + n 2 3 > (-117 - 231 n - 153 n - 34 n ) S + n 2 3 > (1 + 3 n + 3 n + n )} In[46]:= First[CreativeTelescoping[Exp[x]+Exp[y]/Sqrt[x+y], Der[y], {Der[x]}]] Out[46]= {D + 1} x In[47]:= First[CreativeTelescoping[(x+y)^2 Exp[x]+Exp[y]/Sqrt[x+y], Der[y], {Der[x]}]] Out[47]= {D + 1} x In[48]:= First[CreativeTelescoping[(x/y+x y)^2 Exp[x]+Exp[y]/Sqrt[x+y], Der[y], {Der[x]}]] Out[48]= {D + 1} x In[49]:= First[CreativeTelescoping[(x/y+x y)^2 Exp[x]+Exp[y^2]/Sqrt[x+y], Der[y], {Der[x]}]] 2 Out[49]= {D - 2 x D - 1} x x In[50]:= Integrate[Exp[-x^2-y^2], {x, -Pi, Pi}] Sqrt[Pi] Erf[Pi] Out[50]= ---------------- 2 y E In[51]:= Annihilator[%] Out[51]= {D + 2 y} y In[52]:= CreativeTelescoping[Exp[-x^2-y^2], Der[x], {Der[y]}] Out[52]= {{D + 2 y}, {0}} y In[53]:=