A172158 Number of ways to place 6 nonattacking kings on an n X n board.
0, 0, 0, 0, 978, 62266, 1220298, 12033330, 77784658, 377818258, 1492665418, 5042436754, 15062292834, 40736208186, 101489568538, 235984235970, 517314078210, 1077720399538, 2147500025914, 4114538426818, 7613150953522, 13653752767866, 23808409699242
Offset: 1
Links
- Vincenzo Librandi, 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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78, -13,1).
Programs
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Mathematica
CoefficientList[Series[2*x^4*(489 +24776*x +243562*x^2 +373248*x^3 -287097*x^4 -263140*x^5 +376992*x^6 -162056*x^7 +36103*x^8 -20892*x^9 +14622*x^10 -4432*x^11 +465*x^12)/(1-x)^13, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *) -
SageMath
[0,0,0,0]+[(n^12 -135*n^10 +180*n^9 +7465*n^8 -18840*n^7 -202665*n^6 + 751860*n^5 +2442334*n^4 -13441200*n^3 -3643800*n^2 +89860320*n -108217440)/720 for n in (5..40)] # G. C. Greubel, Apr 21 2022
Formula
a(n) = (n^12 - 135*n^10 + 180*n^9 + 7465*n^8 - 18840*n^7 - 202665*n^6 + 751860*n^5 + 2442334*n^4 - 13441200*n^3 - 3643800*n^2 + 89860320*n - 108217440)/720, n>=5. For any fixed value of k > 1, a(n) = n^(2*k)/k! - 9*n^(2*k-2)/2/(k-2)! + 6*n^(2*k-3)/(k-2)! ... - Vaclav Kotesovec, Jan 27 2010
G.f.: 2*x^5 * (489 + 24776*x + 243562*x^2 + 373248*x^3 - 287097*x^4 - 263140*x^5 + 376992*x^6 - 162056*x^7 + 36103*x^8 - 20892*x^9 + 14622*x^10 - 4432*x^11 + 465*x^12)/(1-x)^13. - Vaclav Kotesovec, Mar 24 2010
Extensions
More terms from Vincenzo Librandi, May 27 2013