A179426 Number of ways to place 6 nonattacking kings on an n X n toroidal board.
0, 0, 0, 0, 0, 10596, 486668, 7063520, 55345356, 299491100, 1263811604, 4455716184, 13701863604, 37823872044, 95648273100, 224887404416, 497181121100, 1042609380588, 2088337713332, 4017815773400, 7459198321428, 13414493857116, 23444476061772, 39928736913120, 66425550447500, 108162598959740, 172697249542932, 270794133842456, 417578468928308, 634036069773900
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- V. 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[4 x^5 (426 x^13 - 4263 x^12 + 22311 x^11 - 82449 x^10 + 220918 x^9 - 391803 x^8 + 369356 x^7 + 10716 x^6 - 382230 x^5 + 163719 x^4 + 387689 x^3 - 390831 x^2 - 87230 x - 2649) / (x - 1)^13, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)
Formula
a(n) = 1/720*n^2*(n^10 -135*n^8 +7525*n^6 -217665*n^4 +3289354*n^2 -20949480), n>=7.
G.f.: 4*x^6*(426*x^13 - 4263*x^12 + 22311*x^11 - 82449*x^10 + 220918*x^9 - 391803*x^8 + 369356*x^7 + 10716*x^6 - 382230*x^5 + 163719*x^4 + 387689*x^3 - 390831*x^2 - 87230*x - 2649)/(x-1)^13.