A243719 Number of inequivalent (mod D_8) ways to place 4 nonattacking knights on an n X n board.
1, 6, 66, 609, 3375, 14181, 47485, 136085, 342739, 784059, 1653033, 3267471, 6107271, 10901405, 18683285, 30934341, 49659915, 77611995, 118386689, 176753639, 258774303, 372270981, 526962861, 735113445, 1011678595, 1375177451, 1847843545, 2456771055, 3234056439
Offset: 2
Links
- Heinrich Ludwig, Table of n, a(n) for n = 2..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1).
Programs
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Magma
[1,6,66,609] cat [(n^8 - 54*n^6 + 144*n^5 + 1048*n^4 - 5280*n^3 - 2432*n^2 + 52800*n - 78912 + (1 - (-1)^n)/2*(14*n^4 - 48*n^3 - 158*n^2 + 768*n - 723))/192: n in [6..30]]; // Vincenzo Librandi, Jun 21 2014
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Mathematica
Drop[CoefficientList[Series[411 + 171*x + 38*x^2 - 5*x^3 - 15*x^4 - 6*x^5 - (411 - 1473*x - 236*x^2 + 6588*x^3 - 5073*x^4 - 11179*x^5 + 13200*x^6 + 4572*x^7 - 19047*x^8 - 991*x^9 + 9564*x^10 - 1776*x^11 - 1955*x^12 + 675*x^13) / ((1-x)^9*(1+x)^5), {x, 0, 20}], x],2] (* Vaclav Kotesovec, Jun 19 2014 *)
Formula
a(n) = (n^8 - 54*n^6 + 144*n^5 + 1048*n^4 - 5280*n^3 - 2432*n^2 + 52800*n - 78912 + (1 - (-1)^n)/2*(14*n^4 - 48*n^3 - 158*n^2 + 768*n - 723))/192 for n >= 6.
G.f.: 411 + 171*x + 38*x^2 - 5*x^3 - 15*x^4 - 6*x^5 - (411 - 1473*x - 236*x^2 + 6588*x^3 - 5073*x^4 - 11179*x^5 + 13200*x^6 + 4572*x^7 - 19047*x^8 - 991*x^9 + 9564*x^10 - 1776*x^11 - 1955*x^12 + 675*x^13) / ((1-x)^9*(1+x)^5). - Vaclav Kotesovec, Jun 19 2014