A323501 - OEIS

%I #16 Feb 16 2025 08:33:57

%S 2,6,5,2,22,356,108,24,672,25056,4680,720,38160,2531520,342720,40320,

%T 3467520,358444800,38102400,3628800,460857600,68388364800,5987520000,

%U 479001600,84304281600,16979648716800,1264085222400,87178291200,20312541849600

%N Number of minimum dominating sets in the n X n white bishop graph.

%H Andrew Howroyd, <a href="/A323501/b323501.txt">Table of n, a(n) for n = 2..50</a>

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

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

%F From _Andrew Howroyd_, Sep 09 2019: (Start)

%F a(n) = (n/2)! * (n + 1)/2 for n mod 4 = 0;

%F a(n) = ((n-1)/2)! for n mod 4 = 1;

%F a(n) = (n/2-1)! * (n^2 + n + 2)/4 for n mod 4 = 2;

%F a(n) = ((n-3)/2)! * (n + 1)*(n^3 + n^2 - 6*n + 6)/16 for n mod 4 = 3.

%F (End)

%o (PARI) \\ See A289170 for DomSetCount, Bishop.

%o a(n)={Vec(DomSetCount(Bishop(n, 1), x + O(x^((n+3)\2))))[1]} \\ _Andrew Howroyd_, Sep 08 2019

%o (PARI) a(n)=(n\4*2)!*if(n%4<2, if(n%2==0, (n + 1)/2, 1), if(n%2==0, (n^2 + n + 2)/4, (n + 1)*(n^3 + n^2 - 6*n + 6)/16)); \\ _Andrew Howroyd_, Sep 09 2019

%Y Cf. A182333 (bishop graph), A323500 (black bishop graph).

%Y Cf. A287897, A289170, A303144.

%K nonn,changed

%O 2,1

%A _Eric W. Weisstein_, Jan 16 2019

%E Offset corrected and terms a(11) and beyond from _Andrew Howroyd_, Sep 08 2019