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A182333 Number of arrangements of n bishops such that every square of the board is controlled by at least one bishop. 7
1, 4, 6, 25, 104, 484, 2136, 11664, 71136, 451584, 3006720, 21902400, 176774400, 1456185600, 12758860800, 117456998400, 1181072793600, 12023694950400, 130072449024000, 1451792885760000, 17487355576320000, 212389727477760000, 2729844680048640000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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

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}]

PROG

(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

CROSSREFS

Cf. A005635, A122749, A323500, A323501.

Sequence in context: A272306 A294996 A176858 * A028273 A024471 A075277

Adjacent sequences:  A182330 A182331 A182332 * A182334 A182335 A182336

KEYWORD

nonn,changed

AUTHOR

Vaclav Kotesovec, Apr 25 2012

STATUS

approved

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Last modified September 19 12:57 EDT 2019. Contains 327198 sequences. (Running on oeis4.)