A303111 -id:A303111 - cp's OEIS frontend

A303118 Array read by antidiagonals: T(m,n) = number of minimal total dominating sets in the grid graph P_m X P_n.

0, 1, 1, 2, 4, 2, 1, 4, 4, 1, 2, 16, 6, 16, 2, 4, 16, 49, 49, 16, 4, 3, 49, 66, 169, 66, 49, 3, 4, 81, 225, 576, 576, 225, 81, 4, 8, 169, 640, 2601, 2622, 2601, 640, 169, 8, 9, 324, 1681, 10000, 14400, 14400, 10000, 1681, 324, 9, 10, 625, 4641, 38416, 81055, 137641, 81055, 38416, 4641, 625, 10

Offset: 1

Andrew Howroyd, Apr 18 2018

			Table begins:
=============================================================
m\n| 1   2    3     4      5       6         7          8
---+---------------------------------------------------------
1  | 0   1    2     1      2       4         3          4 ...
2  | 1   4    4    16     16      49        81        169 ...
3  | 2   4    6    49     66     225       640       1681 ...
4  | 1  16   49   169    576    2601     10000      38416 ...
5  | 2  16   66   576   2622   14400     81055     440896 ...
6  | 4  49  225  2601  14400  137641   1081600    8185321 ...
7  | 3  81  640 10000  81055 1081600  11458758  125955729 ...
8  | 4 169 1681 38416 440896 8185321 125955729 1944369025 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..435 (first 29 antidiagonals)
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.

Rows 1..2 are A302655, A303072.
Main diagonal is A303161.
Cf. A300358, A303111, A303293.

A303293 Array read by antidiagonals: T(m,n) = number of minimum total dominating sets in the grid graph P_m X P_n.

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 1, 1, 1, 1, 1, 16, 2, 16, 1, 4, 9, 1, 1, 9, 4, 3, 1, 3, 16, 3, 1, 3, 1, 64, 4, 256, 256, 4, 64, 1, 2, 16, 4, 4, 160, 4, 4, 16, 2, 9, 1, 9, 121, 25, 25, 121, 9, 1, 9, 4, 169, 12, 2916, 268, 144, 268, 2916, 12, 169, 4

Offset: 1

Andrew Howroyd, Apr 20 2018

The minimum size of a total dominating set is the total domination number A300358(m, n).

			Table begins:
===============================================
m\n| 1  2  3    4    5     6   7    8     9
---+-------------------------------------------
1  | 0  1  2    1    1     4   3    1     2 ...
2  | 1  4  1   16    9     1  64   16     1 ...
3  | 2  1  2    1    3     4   4    9    12 ...
4  | 1 16  1   16  256     4 121 2916    25 ...
5  | 1  9  3  256  160    25 268 4225   510 ...
6  | 4  1  4    4   25   144 529 2025 10404 ...
7  | 3 64  4  121  268   529   4  441   630 ...
8  | 1 16  9 2916 4225  2025 441  256     9 ...
9  | 2  1 12   25  510 10404 630    9  1364 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..435 (first 29 antidiagonals)
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.

Rows 1..2 are A302654, A303054.
Main diagonal is A303142.
Cf. A300358, A303111, A303118.

A300358 Array read by antidiagonals: T(m,n) = total domination number of the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 5, 4, 4, 4, 6, 6, 8, 8, 6, 6, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 6, 6, 8, 10, 10, 10, 10, 8, 6, 6, 6, 8, 9, 12, 12, 12, 12, 12, 9, 8, 6, 6, 8, 10, 12, 14, 14, 14, 14, 12, 10, 8, 6, 7, 8, 11, 14, 15, 16, 15, 16, 15, 14, 11, 8, 7

Offset: 1

Andrew Howroyd, Apr 20 2018

			Table begins:
=======================================================
m\n| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
---+---------------------------------------------------
1  | 1  2  2  2  3  4  4  4  5  6  6  6  7  8  8  8 ...
2  | 2  2  2  4  4  4  6  6  6  8  8  8 10 10 10 12 ...
3  | 2  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...
4  | 2  4  4  6  8  8 10 12 12 14 14 16 18 18 20 20 ...
5  | 3  4  5  8  9 10 12 14 15 16 18 20 21 22 24 26 ...
6  | 4  4  6  8 10 12 14 16 18 20 20 24 24 26 28 30 ...
7  | 4  6  7 10 12 14 15 18 20 22 24 26 27 30 32 34 ...
8  | 4  6  8 12 14 16 18 20 22 24 28 30 32 34 36 38 ...
9  | 5  6  9 12 15 18 20 22 25 28 30 32 35 38 40 42 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..435 (first 29 antidiagonals)
Alexandre Talon, Intensive use of computing resources for dominations in grids and other combinatorial problems, arXiv:2002.11615 [cs.DM], 2020. See Sec. 2.3.2.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Total Domination Number

Rows 1..2 are A004524(n+2), A302402.
Main diagonal is A302488.
Cf. A303111, A303118, A303293.

A303114 Array read by antidiagonals: T(m,n) = number of total dominating sets in the n X m king graph.

Original entry on oeis.org

0, 1, 1, 3, 11, 3, 4, 47, 47, 4, 5, 165, 353, 165, 5, 9, 625, 2545, 2545, 625, 9, 16, 2435, 19651, 35458, 19651, 2435, 16, 25, 9367, 150719, 538977, 538977, 150719, 9367, 25, 39, 35901, 1149593, 8213971, 16322279, 8213971, 1149593, 35901, 39

Offset: 1

Andrew Howroyd, Apr 18 2018

			Table begins:
============================================================================
m\n|  1    2       3         4           5             6               7
---|------------------------------------------------------------------------
1  |  0    1       3         4           5             9              16 ...
2  |  1   11      47       165         625          2435            9367 ...
3  |  3   47     353      2545       19651        150719         1149593 ...
4  |  4  165    2545     35458      538977       8213971       124153394 ...
5  |  5  625   19651    538977    16322279     496873689     14980146565 ...
6  |  9 2435  150719   8213971   496873689   30158547693   1812834702647 ...
7  | 16 9367 1149593 124153394 14980146565 1812834702647 217221533288240 ...
...

Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Total Dominating Set

Rows 1..2 are A195971(n-1), A219079.
Main diagonal is A303116.
Cf. A218663 (dominating sets), A291873 (connected dominating sets).
Cf. A303111 (grid graph).

A133793 Number of n X n binary matrices with every element adjacent to some 0 horizontally or vertically.

Original entry on oeis.org

0, 9, 161, 11236, 3270375, 3416753209, 13057656650476, 183641258024619409, 9467975365082755623313, 1790666611784474236947695716, 1242439796942800308861023912760955, 3162387266575455438028580027523664208409, 29528027257213781096968410946112929405583355140

Offset: 1

R. H. Hardin, Jan 05 2008

nonn

a(n) is the number of total dominating sets of the n X n grid graph. - Eric W. Weisstein, Apr 19 2018

Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Total Dominating Set

Main diagonal of A303111.

a(1) = 0 prepended by Eric W. Weisstein, Apr 19 2018

a(12)-a(13) from Andrew Howroyd, Apr 19 2018

cp's OEIS Frontend

A303118 Array read by antidiagonals: T(m,n) = number of minimal total dominating sets in the grid graph P_m X P_n.

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A303293 Array read by antidiagonals: T(m,n) = number of minimum total dominating sets in the grid graph P_m X P_n.

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A300358 Array read by antidiagonals: T(m,n) = total domination number of the grid graph P_m X P_n.

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A303114 Array read by antidiagonals: T(m,n) = number of total dominating sets in the n X m king graph.

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A133793 Number of n X n binary matrices with every element adjacent to some 0 horizontally or vertically.

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