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

1,7

Comments

An amazon (superqueen) moves like a queen and a knight.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

V. Kotesovec, Non-attacking chess pieces, 6ed, 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).

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 *)

Crossrefs

Cf. A173214, A172519, A178972, A178973.

Sequence in context: A188484 A271037 A243477 * A233920 A017814 A035744

Adjacent sequences: A178971 A178972 A178973 * A178975 A178976 A178977

Keyword

nonn,easy

Author

Vaclav Kotesovec, Jan 02 2011

Status

approved