A173775 Number of ways to place 5 nonattacking queens on an n X n toroidal board

0, 0, 0, 0, 10, 0, 882, 13312, 85536, 561440, 2276736, 9471744, 27991470, 85725696, 209107890, 525062144, 1116665944, 2437807104, 4691672964, 9234168960, 16462896030, 29919532544, 50215537658, 85687824384, 136944081500

Offset

1,5

Links

Table of n, a(n) for n=1..25.

Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes.

Formula

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.

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).

Crossrefs

Cf. A172517, A172518, A172519, A007705, A108792, A061991.

Sequence in context: A070190 A216797 A221305 * A222401 A287465 A288013

Adjacent sequences: A173772 A173773 A173774 * A173776 A173777 A173778

Keyword

nonn,nice

Author

Vaclav Kotesovec, Feb 24 2010

Status

approved