A061992 Number of ways to place 6 nonattacking queens on a 6 X n board.
0, 0, 0, 0, 0, 0, 4, 94, 550, 2292, 7552, 21362, 52856, 117694, 241484, 463038, 838816, 1448002, 2398292, 3832374, 5935120, 8941514, 13145292, 18908302, 26670584, 36961170, 50409604, 67758182, 89874912, 117767194, 152596220
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes, part of V. Kotesovec, Between chessboard and computer, 1996, pp. 204 - 206.
Programs
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Mathematica
CoefficientList[Series[-2 x^6 (4 x^17 -12 x^16 + 12 x^15 + 10 x^14 - 10 x^13 + 40 x^12 - 278 x^11 + 677 x^10 - 582 x^9 - 62 x^8 + 654 x^7 - 501 x^6 + 293 x^5 - 46 x^4 + 138 x^3 - 12 x^2 + 33 x + 2) / (x-1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)
Formula
G.f.: - 2*x^6*(4*x^17 - 12*x^16 + 12*x^15 + 10*x^14 - 10*x^13 + 40*x^12 - 278*x^11 + 677*x^10 - 582*x^9 - 62*x^8 + 654*x^7 - 501*x^6 + 293*x^5 - 46*x^4 + 138*x^3 - 12*x^2 + 33*x + 2)/(x - 1)^7.
Recurrence: a(n) = 7*a(n - 1) - 21*a(n - 2) + 35*a(n - 3) - 35*a(n - 4) + 21*a(n - 5) - 7*a(n - 6) + a(n - 7), n >= 24.
Explicit formula (V.Kotesovec, 1992): a(n) = n^6 - 45*n^5 + 943*n^4 - 11755*n^3 + 91480*n^2 - 418390*n + 870920, n >= 17.