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%I #21 Sep 12 2015 11:00:23
%S 0,0,0,0,0,0,98,3328,17496,99600,316052,1041408,2501538,6157536,
%T 12531150,25938944,47168268,86938272,145818008,247240000,390084786,
%U 620964256,933865918,1414946304,2047225000,2980849040,4177648224,5886858432,8032809818,11012886000,14689386642,19674427392,25732782504,33779841296,43433208000,56027023488,70963952198,90145026976,112667956362,141187744000
%N Number of ways to place 4 nonattacking amazons (superqueens) on an n X n toroidal board.
%C An amazon (superqueen) moves like a queen and a knight.
%H Vincenzo Librandi, <a href="/A178974/b178974.txt">Table of n, a(n) for n = 1..1000</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013
%F 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.
%F 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).
%t 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 *)
%Y Cf. A173214, A172519, A178972, A178973.
%K nonn,easy
%O 1,7
%A _Vaclav Kotesovec_, Jan 02 2011
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