A182333 - OEIS

%I #27 Sep 10 2019 03:18:51

%S 1,4,6,25,104,484,2136,11664,71136,451584,3006720,21902400,176774400,

%T 1456185600,12758860800,117456998400,1181072793600,12023694950400,

%U 130072449024000,1451792885760000,17487355576320000,212389727477760000,2729844680048640000

%N Number of arrangements of n bishops such that every square of the board is controlled by at least one bishop.

%C Number of minimum dominating sets in the n X n bishop graph. - _Eric W. Weisstein_, Jun 04 2017

%D A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, vol.1, 1987, p.11 and p.83-88.

%H Vincenzo Librandi, <a href="/A182333/b182333.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BishopGraph.html">Bishop Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>

%F a(n) = (((2*floor(n/4))!)^2/128)*(n^5 + 3*n^4 + n^3 + 35*n^2 + 38*n + 2 - (n^5 - n^4 - 7*n^3 - n^2 - 10*n - 30)*(-1)^n -4*(n^3 + 2*n^2 + n - 4)*n*cos(Pi*n/2) - 2*(n^5 + n^4 - 11*n^3 - 7*n^2 - 2*n + 2)*sin(Pi*n/2)), for n > 1.

%F a(n) = A323500(n) * A323501(n) for n > 1. - _Andrew Howroyd_, Sep 08 2019

%t Table[If[n==1,1,((2*Floor[n/4])!)^2/128*(n^5+3*n^4+n^3+35*n^2+38*n+2-(n^5-n^4-7*n^3-n^2-10*n-30)*(-1)^n-4*(n^3+2*n^2+n-4)*n*Cos[Pi*n/2]-2*(n^5+n^4-11*n^3-7*n^2-2*n+2)*Sin[Pi*n/2])],{n,1,25}]

%o (PARI) a(n)={if(n==1, 1, (n\4*2)!^2*if(n%4<2, if(n%2==0, (n+1)^2, (n^3 + 3*n^2 + 2*n - 2)/2), if(n%2==0, (n^2+n+2)^2/4, (n+1)*(n-1)*(n^3 + n^2 - 6*n + 6)/8))/4)} \\ _Andrew Howroyd_, Sep 09 2019

%Y Cf. A005635, A122749, A323500, A323501.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, Apr 25 2012