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A350815
Array read by antidiagonals: T(m,n) is the number of minimum dominating sets in the m X n king graph.
8
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, 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, 208, 1, 1024, 1656, 1, 1656, 1024, 1, 208, 5, 1, 80, 13, 64, 408, 64, 64, 408, 64, 13, 80, 1
OFFSET
1,2
COMMENTS
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)).
LINKS
Stephan Mertens, Table of n, a(n) for n = 1..946 (first 276 terms from Andrew Howroyd)
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Minimum Dominating Set
FORMULA
T(n,m) = T(m,n).
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.
T(3*m-1, 3*n-1) = A350819(m, n).
EXAMPLE
Table begins:
============================================
m\n | 1 2 3 4 5 6 7 8
----+---------------------------------------
1 | 1 2 1 4 3 1 8 4 ...
2 | 2 4 2 16 12 4 64 32 ...
3 | 1 2 1 4 3 1 8 4 ...
4 | 4 16 4 256 144 16 4096 1024 ...
5 | 3 12 3 144 79 9 1656 408 ...
6 | 1 4 1 16 9 1 64 16 ...
7 | 8 64 8 4096 1656 64 243856 29744 ...
8 | 4 32 4 1024 408 16 29744 3600 ...
...
CROSSREFS
Rows 1..3 are A347633, A350816, A347633.
Main diagonal is A347554.
Cf. A075561, A218663 (dominating sets), A286849 (minimal dominating sets), A303335, A350818, A350819.
Sequence in context: A061298 A276468 A002126 * A129721 A268193 A238606
KEYWORD
nonn,look,tabl
AUTHOR
Andrew Howroyd, Jan 17 2022
STATUS
approved