A061997 Number of ways to place 4 nonattacking kings on an n X n board.
0, 0, 0, 1, 79, 1987, 16834, 85275, 317471, 962089, 2515262, 5882109, 12605095, 25175191, 47443474, 85152487, 146608359, 243516365, 392004286, 613859609, 938008287, 1402264459, 2055382210, 2959442131, 4192607119
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.
- Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
Programs
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Magma
R
:=PowerSeriesRing(Integers(), 50); [0,0,0] cat Coefficients(R!( x^3*(1 +70*x +1312*x^2 +1711*x^3 -1209*x^4 -1060*x^5 +1186*x^6 -361*x^7 +30*x^8)/(1-x)^9 )); // G. C. Greubel, Apr 30 2022 -
Mathematica
CoefficientList[Series[x^3*(1 +70*x +1312*x^2 +1711*x^3 -1209*x^4 -1060*x^5 +1186*x^6 -361*x^7 +30*x^8)/(1-x)^9, {x, 0, 50}], x] (* Vincenzo Librandi, May 02 2013 *) -
SageMath
[0,0,0]+[(n^8 -54*n^6 +72*n^5 +995*n^4 -2472*n^3 -5094*n^2 +21480*n -17112)/24 for n in (3..50)] # G. C. Greubel, Apr 30 2022
Formula
G.f.: x^3*(1 + 70*x + 1312*x^2 + 1711*x^3 - 1209*x^4 - 1060*x^5 + 1186*x^6 - 361*x^7 + 30*x^8)/(1 - x)^9.
Recurrence: a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9), n >= 12.
Explicit formula (K.Fabel and K.Soltsien): a(n) = (n^8 - 54*n^6 + 72*n^5 + 995*n^4 - 2472*n^3 - 5094*n^2 + 21480*n - 17112)/24, n >= 3.
a(n) = A193580(n,4). - R. J. Mathar, Sep 03 2016