A182333 Number of arrangements of n bishops such that every square of the board is controlled by at least one bishop.

1, 4, 6, 25, 104, 484, 2136, 11664, 71136, 451584, 3006720, 21902400, 176774400, 1456185600, 12758860800, 117456998400, 1181072793600, 12023694950400, 130072449024000, 1451792885760000, 17487355576320000, 212389727477760000, 2729844680048640000

Offset: 1

Author

Vaclav Kotesovec, Apr 25 2012

Keywords

• nonn

Comments

Number of minimum dominating sets in the n X n bishop graph. - Eric W. Weisstein, Jun 04 2017

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Bishop Graph
• Eric Weisstein's World of Mathematics, Dominating Set

Crossrefs

Cf. A005635, A122749, A323500, A323501.

Programs

Mathematica

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}]

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

Formula

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.

a(n) = A323500(n) * A323501(n) for n > 1. - Andrew Howroyd, Sep 08 2019