A172519 Number of ways to place 4 nonattacking queens on an n X n toroidal board.

0, 0, 0, 0, 50, 288, 2450, 16384, 62208, 233600, 638880, 1755072, 3901534, 8772176, 17051850, 33507328, 59175640, 105557904, 173570244, 287904000, 447885774, 702042000, 1044894554, 1565385984, 2247132500, 3244194304, 4519015596

Offset

1,5

Links

Jinyuan Wang, Table of n, a(n) for n = 1..1000

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

Index entries for linear recurrences with constant coefficients, signature (-1, 3, 6, 3, -9, -20, -11, 15, 40, 31, -15, -53, -50, 0, 50, 53, 15, -31, -40, -15, 11, 20, 9, -3, -6, -3, 1, 1).

Formula

a(n) = (n^8/24 - n^7 + 245n^6/24 - 113n^5/2 + 2843n^4/16 - 593n^3/2 + 4757n^2/24) + (n^6/8 - 5n^5/2 + 305n^4/16 - 129n^3/2 + 629n^2/8)*(-1)^n + 8n^2*cos(2*Pi*n/3)/3 + 9n^2*cos(Pi*n/2)/2.

Recurrence: a(n) = -a(n-1) + 3a(n-2) + 6a(n-3) + 3a(n-4) - 9a(n-5) - 20a(n-6) - 11a(n-7) + 15a(n-8) + 40a(n-9) + 31a(n-10) - 15a(n-11) - 53a(n-12) - 50a(n-13) + 50a(n-15) + 53a(n-16) + 15a(n-17) - 31a(n-18) - 40a(n-19) - 15a(n-20) + 11a(n-21) + 20a(n-22) + 9a(n-23) - 3a(n-24) - 6a(n-25) - 3a(n-26) + a(n-27) + a(n-28), n >= 29. - Vaclav Kotesovec, Feb 09 2010

G.f.: -2*x^5*(287*x^22 + 5191*x^21 + 25616*x^20 + 105043*x^19 + 280800*x^18 + 651461*x^17 + 1186795*x^16 + 1925172*x^15 + 2611064*x^14 + 3190574*x^13 + 3337574*x^12 + 3161250*x^11 + 2574658*x^10 + 1891298*x^9 + 1175308*x^8 + 649556*x^7 + 291897*x^6 + 115771*x^5 + 34682*x^4 + 8835*x^3 + 1294*x^2 + 169*x + 25) / ((x - 1)^9*(x + 1)^7*(x^2 + 1)^3*(x^2 + x + 1)^3). - Colin Barker, Sep 21 2014

Mathematica

LinearRecurrence[{-1, 3, 6, 3, -9, -20, -11, 15, 40, 31, -15, -53, -50, 0, 50, 53, 15, -31, -40, -15, 11, 20, 9, -3, -6, -3, 1, 1}, {0, 0, 0, 0, 0, 50, 288, 2450, 16384, 62208, 233600, 638880, 1755072, 3901534, 8772176, 17051850, 33507328, 59175640, 105557904, 173570244, 287904000, 447885774, 702042000, 1044894554, 1565385984, 2247132500, 3244194304, 4519015596}, 40] (* Harvey P. Dale, Apr 30 2018 *)

Crossrefs

Cf. A007705, A061994, A172517, A172518, A173775, A178722.

Sequence in context: A186123 A220864 A205355 * A046656 A205620 A183698

Adjacent sequences: A172516 A172517 A172518 * A172520 A172521 A172522

Keyword

nonn,easy

Author

Vaclav Kotesovec, Feb 05 2010

Status

approved