A392413 Array read by antidiagonals: T(m,n) is the number of partitions of the vertices of the grid graph P_m X P_n into total dominating sets.

0, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 9, 15, 9, 1, 1, 19, 33, 33, 19, 1, 1, 51, 187, 339, 187, 51, 1, 1, 129, 723, 2313, 2313, 723, 129, 1, 1, 339, 3265, 20001, 36427, 20001, 3265, 339, 1, 1, 883, 14451, 155683, 498003, 498003, 155683, 14451, 883, 1, 1, 2313, 63707, 1267233, 7322241, 13096963, 7322241, 1267233, 63707, 2313, 1

Offset

1,5

Comments

The vertices of a grid graph can be partitioned into at most 2 parts such that each part is a total dominating set.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..231 (first 21 antidiagonals)

Eric Weisstein's World of Mathematics, Grid Graph.

Eric Weisstein's World of Mathematics, Total Dominating Set.

Formula

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

T(m,n) = A203285(m,n)/2 + 1, with T(1,1) = 0.

Example

Array begins:
=========================================================
n\k | 1   2    3      4       5         6           7 ...
----+----------------------------------------------------
  1 | 0   1    1      1       1         1           1 ...
  2 | 1   3    3      9      19        51         129 ...
  3 | 1   3   15     33     187       723        3265 ...
  4 | 1   9   33    339    2313     20001      155683 ...
  5 | 1  19  187   2313   36427    498003     7322241 ...
  6 | 1  51  723  20001  498003  13096963   337376289 ...
  7 | 1 129 3265 155683 7322241 337376289 15873640451 ...
  ...

Crossrefs

Main diagonal is A392414.

Rows 2..3 are A392415, A392416.

Cf. A203285, A303111, A391824.

Sequence in context: A134444 A176149 A091442 * A379473 A025834 A257379

Adjacent sequences: A392410 A392411 A392412 * A392414 A392415 A392416

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 10 2026

Status

approved