cp's OEIS Frontend

A061997 Number of ways to place 4 nonattacking kings on an n X n board.

%I A061997 #18 Apr 30 2022 01:35:58
%S A061997 0,0,0,1,79,1987,16834,85275,317471,962089,2515262,5882109,12605095,
%T A061997 25175191,47443474,85152487,146608359,243516365,392004286,613859609,
%U A061997 938008287,1402264459,2055382210,2959442131,4192607119
%N A061997 Number of ways to place 4 nonattacking kings on an n X n board.
%H A061997 Vincenzo Librandi, <a href="/A061997/b061997.txt">Table of n, a(n) for n = 0..1000</a>
%H A061997 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>, part of V. Kotesovec, Between chessboard and computer, 1996, pp. 204 - 206.
%H A061997 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A061997 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.
%F A061997 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.
%F A061997 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.
%F A061997 a(n) = A193580(n,4). - _R. J. Mathar_, Sep 03 2016
%t A061997 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 *)
%o A061997 (Magma) R<x>:=PowerSeriesRing(Integers(), 50);
%o A061997 [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
%o A061997 (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
%Y A061997 Cf. A061995, A061996, A061998, A193580.
%K A061997 nonn,easy
%O A061997 0,5
%A A061997 Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001