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A323500 Number of minimum dominating sets in the n X n black bishop graph.

%I A323500 #21 Feb 27 2025 16:28:49
%S A323500 1,2,1,5,52,22,6,108,2964,672,120,4680,245520,38160,5040,342720,
%T A323500 29292480,3467520,362880,38102400,4819046400,460857600,39916800,
%U A323500 5987520000,1050690009600,84304281600,6227020800,1264085222400,293878019635200,20312541849600
%N A323500 Number of minimum dominating sets in the n X n black bishop graph.
%H A323500 Andrew Howroyd, <a href="/A323500/b323500.txt">Table of n, a(n) for n = 1..50</a>
%H A323500 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BlackBishopGraph.html">Black Bishop Graph</a>
%H A323500 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumDominatingSet.html">Minimum Dominating Set</a>
%F A323500 From _Andrew Howroyd_, Sep 09 2019: (Start)
%F A323500 a(n) = (n/2)! * (n + 1)/2 for n mod 4 = 0;
%F A323500 a(n) = ((n-1)/2)! * (n^3 + 3*n^2 + 2*n - 2)/8 for n mod 4 = 1, n > 1;
%F A323500 a(n) = (n/2-1)! * (n^2 + n + 2)/4 for n mod 4 = 2;
%F A323500 a(n) = ((n-1)/2)! for n mod 4 = 3.
%F A323500 (End)
%t A323500 Table[Piecewise[{{1, n == 1}, {(n/2)! (n + 1)/2, Mod[n, 4] == 0}, {((n - 1)/2)! (n^3 + 3 n^2 + 2 n - 2)/8, Mod[n, 4] == 1}, {(n/2 - 1)! (n^2 + n + 2)/4, Mod[n, 4] == 2}, {((n - 1)/2)!, Mod[n, 4] == 3}}], {n, 20}] (* _Eric W. Weisstein_, Feb 27 2025 *)
%o A323500 (PARI) \\ See A286886 for DomSetCount, Bishop.
%o A323500 a(n)={Vec(DomSetCount(Bishop(n, 0), x + O(x^((n+3)\2))))[1]} \\ _Andrew Howroyd_, Sep 08 2019
%o A323500 (PARI) a(n)=if(n==1, 1, (n\4*2)!*if(n%4<2, if(n%2==0, (n+1)/2, (n^3 + 3*n^2 + 2*n - 2)/8), if(n%2==0, (n^2+n+2)/4, (n-1)/2))); \\ _Andrew Howroyd_, Sep 09 2019
%Y A323500 Cf. A182333 (bishop graph), A323501 (white bishop graph).
%Y A323500 Cf. A286886, A289164, A303141.
%K A323500 nonn
%O A323500 1,2
%A A323500 _Eric W. Weisstein_, Jan 16 2019
%E A323500 Terms a(11) and beyond from _Andrew Howroyd_, Sep 08 2019