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A243719 Number of inequivalent (mod D_8) ways to place 4 nonattacking knights on an n X n board. 5

%I #14 Sep 08 2022 08:46:08

%S 1,6,66,609,3375,14181,47485,136085,342739,784059,1653033,3267471,

%T 6107271,10901405,18683285,30934341,49659915,77611995,118386689,

%U 176753639,258774303,372270981,526962861,735113445,1011678595,1375177451,1847843545,2456771055,3234056439

%N Number of inequivalent (mod D_8) ways to place 4 nonattacking knights on an n X n board.

%H Heinrich Ludwig, <a href="/A243719/b243719.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1).

%F 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.

%F 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

%t 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 *)

%o (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

%Y Cf. A243716, A172135, A243717, A243718, A243720.

%K nonn,easy

%O 2,2

%A _Heinrich Ludwig_, Jun 19 2014

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Last modified February 24 04:32 EST 2024. Contains 370288 sequences. (Running on oeis4.)