%I #17 Aug 18 2024 18:48:54

%S 1,2,2,1,4,1,4,2,2,4,3,16,1,16,3,1,12,4,4,12,1,8,4,3,256,3,4,8,4,64,1,

%T 144,144,1,64,4,1,32,8,16,79,16,8,32,1,13,8,4,4096,9,9,4096,4,8,13,5,

%U 208,1,1024,1656,1,1656,1024,1,208,5,1,80,13,64,408,64,64,408,64,13,80,1

%N Array read by antidiagonals: T(m,n) is the number of minimum dominating sets in the m X n king graph.

%C The minimum size of a dominating set is the domination number which in the case of an m X n king graph is given by (ceiling(m/3) * ceiling(n/3)).

%H Stephan Mertens, <a href="/A350815/b350815.txt">Table of n, a(n) for n = 1..946</a> (first 276 terms from Andrew Howroyd)

%H Stephan Mertens, <a href="https://arxiv.org/abs/2408.08053">Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph</a>, arXiv:2408.08053 [math.CO], Aug 2024.

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

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

%F T(n,m) = T(m,n).

%F T(3*m, 3*n) = 1; T(3*m+1, 3*n) = (m^2 + 5*m + 2)^n; T(3*m+2, 3*n) = (m+2)^n.

%F T(3*m-1, 3*n-1) = A350819(m, n).

%e Table begins:

%e ============================================

%e m\n | 1 2 3 4 5 6 7 8

%e ----+---------------------------------------

%e 1 | 1 2 1 4 3 1 8 4 ...

%e 2 | 2 4 2 16 12 4 64 32 ...

%e 3 | 1 2 1 4 3 1 8 4 ...

%e 4 | 4 16 4 256 144 16 4096 1024 ...

%e 5 | 3 12 3 144 79 9 1656 408 ...

%e 6 | 1 4 1 16 9 1 64 16 ...

%e 7 | 8 64 8 4096 1656 64 243856 29744 ...

%e 8 | 4 32 4 1024 408 16 29744 3600 ...

%e ...

%Y Rows 1..3 are A347633, A350816, A347633.

%Y Main diagonal is A347554.

%Y Cf. A075561, A218663 (dominating sets), A286849 (minimal dominating sets), A303335, A350818, A350819.

%K nonn,look,tabl

%O 1,2

%A _Andrew Howroyd_, Jan 17 2022