A061991 Number of ways to place 5 nonattacking queens on a 5 X n board.
0, 0, 0, 0, 0, 10, 40, 164, 568, 1614, 3916, 8492, 16852, 31100, 54068, 89428, 141812, 216932, 321700, 464348, 654548, 903532, 1224212, 1631300, 2141428, 2773268, 3547652, 4487692, 5618900, 6969308, 8569588, 10453172, 12656372, 15218500, 18181988, 21592508
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- V. Kotesovec, Ways of placing non-attacking queens and kings..., part of "Between chessboard and computer", 1996, pp. 204 - 206.
- Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
Programs
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Mathematica
CoefficientList[Series[2 x^5 (4 x^11 -11 x^10 + 16 x^9 + 7 x^8 - 32 x^7 + 38 x^6 + 6 x^5 + 8 x^4 - 8 x^3 + 37 x^2 - 10 x + 5) / (x-1)^6, {x, 0, 30}], x] (* Vincenzo Librandi, May 12 2013 *)
Formula
G.f.: 2*x^5*(4*x^11 - 11*x^10 + 16*x^9 + 7*x^8 - 32*x^7 + 38*x^6 + 6*x^5 + 8*x^4 - 8*x^3 + 37*x^2 - 10*x + 5)/(x - 1)^6.
Recurrence: a(n) = 6*a(n - 1) - 15*a(n - 2) + 20*a(n - 3) - 15*a(n - 4) + 6*a(n - 5) - a(n - 6), n >= 17.
Explicit formula (V. Kotesovec, 1992): a(n) = n^5 - 30*n^4 + 407*n^3 - 3098*n^2 + 13104*n - 24332, n >= 11.