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%I #16 Oct 08 2023 04:53:29
%S 0,0,0,0,10,0,882,13312,85536,561440,2276736,9471744,27991470,
%T 85725696,209107890,525062144,1116665944,2437807104,4691672964,
%U 9234168960,16462896030,29919532544,50215537658,85687824384,136944081500
%N Number of ways to place 5 nonattacking queens on an n X n toroidal board
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>.
%F a(n) = (1/120)*n^10 - (1/3)*n^9 + (143/24)*n^8 - (373/6*n^7) + (99377/240)*n^6 - (3603/2)*n^5 + (119627/24)*n^4 - (23833/3)*n^3 + (16342/3)*n^2 + ((1/24)*n^8 - (3/2)*n^7 + (1111/48)*n^6 - (391/2)*n^5 + (7595/8)*n^4 - 2487*n^3 + (8032/3)*n^2)*(-1)^n + ((9/2)*n^4 - 78*n^3 + 374*n^2)*cos(Pi*n/2) + ((8/3)*n^4 - (128/3)*n^3 + (656/3)*n^2)*cos(2*Pi*n/3) + (80/3)*n^2*cos(Pi*n/3) + (16/5)*n^2*cos(2*Pi*n/5) + (16/5)*n^2*cos(Pi*n/5)*(-1)^n.
%F Recurrence: a(n) = -3a(n-1) - 5a(n-2) - 5a(n-3) + 2a(n-4) + 17a(n-5) + 37a(n-6) + 49a(n-7) + 35a(n-8) - 16a(n-9) - 101a(n-10) - 185a(n-11) - 215a(n-12) - 139a(n-13) + 56a(n-14) + 321a(n-15) + 544a(n-16) + 588a(n-17) + 368a(n-18) - 99a(n-19) - 656a(n-20) - 1069a(n-21) - 1111a(n-22) - 689a(n-23) + 84a(n-24) + 929a(n-25) + 1488a(n-26) + 1506a(n-27) + 939a(n-28) - 939a(n-30) - 1506a(n-31) - 1488a(n-32) - 929a(n-33) - 84a(n-34) + 689a(n-35) + 1111a(n-36) + 1069a(n-37) + 656a(n-38) + 99a(n-39)-368a(n-40) - 588a(n-41) - 544a(n-42) - 321a(n-43) - 56a(n-44) + 139a(n-45) + 215a(n-46) + 185a(n-47) + 101a(n-48) + 16a(n-49) - 35a(n-50) - 49a(n-51) - 37a(n-52) - 17a(n-53) - 2a(n-54) + 5a(n-55) + 5a(n-56) + 3a(n-57) + a(n-58).
%Y Cf. A172517, A172518, A172519, A007705, A108792, A061991.
%K nonn,nice
%O 1,5
%A _Vaclav Kotesovec_, Feb 24 2010
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