A303114 -id:A303114

Search: a303114 -id:a303114

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A303111

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

+10
6

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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

LINKS

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

EXAMPLE

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 ...

...

CROSSREFS

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).

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 18 2018

STATUS

approved

A303335

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

+10
6

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

The minimum size of a total dominating set is the total domination number A303378(m, n).

LINKS

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, Total Dominating Set

EXAMPLE

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 ...

...

CROSSREFS

Rows 1..2 are A302654, A350817.

Main diagonal is A303156.

Cf. A303114, A303293, A303378, A332390, A350815.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 21 2018

STATUS

approved

A332390

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

+10
6

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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, Total Dominating Set

FORMULA

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

EXAMPLE

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 ...

...

CROSSREFS

Rows 1..4 are A302655, A332392, A332393, A332394.

Main diagonal is A332391.

Cf. A286849, A303114, A303118, A303335, A303378.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Feb 10 2020

STATUS

approved

A303378

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

+10
4

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

EXAMPLE

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 ...

...

CROSSREFS

Main diagonal is A302401.

Cf. A300358, A303114, A303335.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 22 2018

STATUS

approved

A303116

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..12.

Eric Weisstein's World of Mathematics, King Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

CROSSREFS

Main diagonal of A303114.

Cf. A133791, A302401, A303156, A332391.

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Apr 18 2018

STATUS

approved

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