A303114 -id:A303114 - cp's OEIS frontend

A303111 Array read by antidiagonals: T(m,n) = number of total dominating sets in the grid graph P_m X P_n.

0, 1, 1, 3, 9, 3, 4, 25, 25, 4, 5, 81, 161, 81, 5, 9, 289, 961, 961, 289, 9, 16, 961, 6235, 11236, 6235, 961, 16, 25, 3249, 39601, 137641, 137641, 39601, 3249, 25, 39, 11025, 251433, 1677025, 3270375, 1677025, 251433, 11025, 39

Offset: 1

Andrew Howroyd, Apr 18 2018

Equivalently, the number of n X m binary matrices with every element adjacent to some 0 horizontally or vertically.

			Table begins:
=======================================================================
m\n|  1    2      3        4          5            6              7
---|-------------------------------------------------------------------
1  |  0    1      3        4          5            9             16 ...
2  |  1    9     25       81        289          961           3249 ...
3  |  3   25    161      961       6235        39601         251433 ...
4  |  4   81    961    11236     137641      1677025       20430400 ...
5  |  5  289   6235   137641    3270375     76405081     1783064069 ...
6  |  9  961  39601  1677025   76405081   3416753209   152598828321 ...
7  | 16 3249 251433 20430400 1783064069 152598828321 13057656650476 ...
...

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

Rows 1..2 are A195971(n-1), A141583(n+1).
Main diagonal is A133793.
Cf. A218354 (dominating sets), A291872 (connected dominating sets).
Cf. A303114 (king graph), A303118 (minimal total dominating sets).

A303335 Array read by antidiagonals: T(m,n) is the number of minimum total dominating sets in the m X n king graph.

Original entry on oeis.org

0, 1, 1, 2, 6, 2, 1, 9, 9, 1, 1, 4, 8, 4, 1, 4, 8, 1, 1, 8, 4, 3, 89, 3, 35, 3, 89, 3, 1, 56, 76, 9, 9, 76, 56, 1, 2, 16, 17, 1, 1, 1, 17, 16, 2, 9, 64, 1, 130, 9, 9, 130, 1, 64, 9, 4, 780, 6, 16, 60, 8684, 60, 16, 6, 780, 4, 1, 304, 229, 1, 89, 493, 493, 89, 1, 229, 304, 1

Offset: 1

Andrew Howroyd, Apr 21 2018

The minimum size of a total dominating set is the total domination number A303378(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  6  9   4  8   89  56  16   64 ...
3  | 2  9  8   1  3   76  17   1    6 ...
4  | 1  4  1  35  9    1 130  16    1 ...
5  | 1  8  3   9  1    9  60  89   45 ...
6  | 4 89 76   1  9 8684 493   1   50 ...
7  | 3 56 17 130 60  493 208  40   32 ...
8  | 1 16  1  16 89    1  40 604    1 ...
9  | 2 64  6   1 45   50  32   1 1192 ...
...

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

Rows 1..2 are A302654, A350817.
Main diagonal is A303156.
Cf. A303114, A303293, A303378, A332390, A350815.

A332390 Array read by antidiagonals: T(m,n) is the number of minimal total dominating sets in the m X n king graph.

Original entry on oeis.org

0, 1, 1, 2, 6, 2, 1, 10, 10, 1, 2, 15, 20, 15, 2, 4, 52, 52, 52, 52, 4, 3, 105, 179, 141, 179, 105, 3, 4, 175, 418, 801, 801, 418, 175, 4, 8, 481, 1167, 2950, 7770, 2950, 1167, 481, 8, 9, 1028, 3498, 9792, 34790, 34790, 9792, 3498, 1028, 9, 10, 2000, 9074, 47527, 184318, 204372, 184318, 47527, 9074, 2000, 10

Offset: 1

Andrew Howroyd, Feb 10 2020

			Array begins:
================================================================
m\n | 1   2    3     4       5        6         7          8
----+-----------------------------------------------------------
  1 | 0   1    2     1       2        4         3          4 ...
  2 | 1   6   10    15      52      105       175        481 ...
  3 | 2  10   20    52     179      418      1167       3498 ...
  4 | 1  15   52   141     801     2950      9792      47527 ...
  5 | 2  52  179   801    7770    34790    184318    1305358 ...
  6 | 4 105  418  2950   34790   204372   1593094   14720683 ...
  7 | 3 175 1167  9792  184318  1593094  16260853  231301551 ...
  8 | 4 481 3498 47527 1305358 14720683 231301551 4570906041 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..120
Eric Weisstein's World of Mathematics, King Graph.
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.

Rows 1..4 are A302655, A332392, A332393, A332394.
Main diagonal is A332391.
Cf. A286849, A303114, A303118, A303335, A303378.

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

A303378 Array read by antidiagonals: T(m,n) = total domination number of the m X n king graph.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 4, 3, 2, 2, 3, 4, 4, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 6, 5, 4, 6, 6, 6, 6, 4, 5, 6, 6, 6, 5, 6, 7, 8, 7, 6, 5, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 6, 6, 6, 6, 7, 6, 6, 8, 9, 8, 9, 8, 9, 8, 6, 6, 7

Offset: 1

Andrew Howroyd, Apr 22 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  2  3  4  4  4  5  6  6  6  7  8  8  8 ...
3  | 2  2  2  2  3  4  4  4  5  6  6  6  7  8  8  8 ...
4  | 2  2  2  4  4  4  6  6  6  8  8  8 10 10 10 12 ...
5  | 3  3  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...
6  | 4  4  4  4  6  8  8  8 10 12 12 12 14 16 16 16 ...
7  | 4  4  4  6  7  8  9 10 11 12 14 14 16 17 18 19 ...
8  | 4  4  4  6  8  8 10 12 12 14 16 16 18 20 20 22 ...
9  | 5  5  5  6  9 10 11 12 15 16 17 18 21 22 23 24 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..496
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Total Dominating Set

Main diagonal is A302401.

Cf. A300358, A303114, A303335.

A303116 Number of total dominating sets in the n X n king graph.

Original entry on oeis.org

0, 11, 353, 35458, 16322279, 30158547693, 217221533288240, 6223220939472363571, 709791800918008570287847, 321673400252458591521699180612, 579292884621843116328602359172702605, 4146239141804826663870561644604700888044071

Offset: 1

Andrew Howroyd, Apr 18 2018

nonn

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

Main diagonal of A303114.

Cf. A133791, A302401, A303156, A332391.

cp's OEIS Frontend

A303111 Array read by antidiagonals: T(m,n) = number of total dominating sets in the grid graph P_m X P_n.

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A303335 Array read by antidiagonals: T(m,n) is the number of minimum total dominating sets in the m X n king graph.

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A332390 Array read by antidiagonals: T(m,n) is the number of minimal total dominating sets in the m X n king graph.

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A303378 Array read by antidiagonals: T(m,n) = total domination number of the m X n king graph.

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A303116 Number of total dominating sets in the n X n king graph.

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