A178974 Number of ways to place 4 nonattacking amazons (superqueens) on an n X n toroidal board.
0, 0, 0, 0, 0, 0, 98, 3328, 17496, 99600, 316052, 1041408, 2501538, 6157536, 12531150, 25938944, 47168268, 86938272, 145818008, 247240000, 390084786, 620964256, 933865918, 1414946304, 2047225000, 2980849040, 4177648224, 5886858432, 8032809818, 11012886000, 14689386642, 19674427392, 25732782504, 33779841296, 43433208000, 56027023488, 70963952198, 90145026976, 112667956362, 141187744000
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- V. Kotesovec, Non-attacking chess pieces, 6ed, 2013
Programs
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Mathematica
CoefficientList[Series[2 x^6 (162 x^30 - 350 x^29 - 1488 x^28 - 718 x^27 + 2389 x^26 + 6635 x^25 + 6157 x^24 - 3372 x^23 - 15873 x^22 - 22215 x^21 - 8561 x^20 + 23622 x^19 + 55919 x^18 + 38469 x^17 - 91949 x^16 - 461696 x^15 - 1076702 x^14 - 1978832 x^13 - 2858196 x^12 - 3576618 x^11 - 3727323 x^10 - 3419559 x^9 - 2634463 x^8 - 1782420 x^7 - 988307 x^6 - 472291 x^5 - 171451 x^4 - 53262 x^3 - 10265 x^2 - 1713 x - 49) / ((x - 1)^9 (x + 1)^7 (x^2 + 1)^3 (x^2 + x + 1)^3), {x, 0, 40}], x] (* _Vincenzo Librandi Jun 01 2013 *)
Formula
a(n)= (1/4)*n^2*(n^6/6 -4*n^5 +197*n^4/6 -66*n^3 -1941*n^2/4 +2638*n -18907/6 +(n^4/2 -10*n^3 +289*n^2/4 -210*n +357/2)*(-1)^n +18*cos(Pi*n/2) +32/3*cos(4*Pi*n/3)), n>=10.
G.f.: 2*x^7*(162*x^30 -350*x^29 -1488*x^28 -718*x^27 +2389*x^26 +6635*x^25 +6157*x^24 -3372*x^23 -15873*x^22 -22215*x^21 -8561*x^20 +23622*x^19 +55919*x^18 +38469*x^17 -91949*x^16 -461696*x^15 -1076702*x^14 -1978832*x^13 -2858196*x^12 -3576618*x^11 -3727323*x^10 -3419559*x^9 -2634463*x^8 -1782420*x^7 -988307*x^6 -472291*x^5 -171451*x^4 -53262*x^3 -10265*x^2 -1713*x -49)/((x-1)^9*(x+1)^7*(x^2+1)^3*(x^2+x+1)^3).
Comments