A350815 Array read by antidiagonals: T(m,n) is the number of minimum dominating sets in the m X n king graph.

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

Andrew Howroyd, Table of n, a(n) for n = 1..276

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

Adjacent sequences: A350812 A350813 A350814 * A350816 A350817 A350818

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 17 2022

Status

approved