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
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).
Programs
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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 *)
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