A332390 -id:A332390 - cp's OEIS frontend

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

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.

A332391 Number of minimal total dominating sets in the n X n king graph.

Original entry on oeis.org

0, 6, 20, 141, 7770, 204372, 16260853, 4570906041, 1437642613959, 1151580438927120

Offset: 1

Andrew Howroyd, Feb 10 2020

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

Main diagonal of A332390.

Cf. A302401, A303116, A303156.

a(9)-a(10) from Christian Sievers, Dec 02 2023

A332392 Number of minimal total dominating sets in the 2 X n king graph.

Original entry on oeis.org

1, 1, 6, 10, 15, 52, 105, 175, 481, 1028, 2000, 4821, 10368, 21285, 48625, 105889, 224578, 498106, 1084331, 2331760, 5121797, 11162455, 24152665, 52751528, 114914016, 249490481, 543949956, 1184591233, 2575134753, 5608825345, 12212701582, 26569155354, 57849402071

Offset: 0

Andrew Howroyd, Feb 10 2020

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, King Graph.
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.
Index entries for linear recurrences with constant coefficients, signature (0,1,5,4,8,16,-8,-48,-64).

Row n=2 of A332390.

Vec((1 + x + 5*x^2 + 4*x^3 - 8*x^6 - 48*x^7 - 64*x^8)/(1 - x^2 - 5*x^3 - 4*x^4 - 8*x^5 - 16*x^6 + 8*x^7 + 48*x^8 + 64*x^9) + O(x^40))

a(n) = a(n-2) + 5*a(n-3) + 4*a(n-4) + 8*a(n-5) + 16*a(n-6) - 8*a(n-7) - 48*a(n-8) - 64*a(n-9) for n >= 9.

G.f.: (1 + x + 5*x^2 + 4*x^3 - 8*x^6 - 48*x^7 - 64*x^8)/(1 - x^2 - 5*x^3 - 4*x^4 - 8*x^5 - 16*x^6 + 8*x^7 + 48*x^8 + 64*x^9).

A332393 Number of minimal total dominating sets in the 3 X n king graph.

Original entry on oeis.org

1, 2, 10, 20, 52, 179, 418, 1167, 3498, 9074, 25737, 72785, 196688, 553939, 1545841, 4257959, 11909122, 33092586, 91740852, 255765150, 710642687, 1974282820, 5494583699, 15269944796, 42450687178, 118071427215, 328208378089, 912542015410, 2537483145585, 7054481803595

Offset: 0

Andrew Howroyd, Feb 10 2020

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row n=3 of A332390.

G.f.: (1 + x + 6*x^2 - x^3 - 2*x^4 - 2*x^5 - 66*x^6 - 124*x^7 - 62*x^8 + 52*x^9 + 13*x^10 + x^11 + 22*x^12 + 203*x^13 + 174*x^14 - 18*x^15 + 12*x^16 + 112*x^17 + 15*x^18 - 161*x^19 - 78*x^20 - 61*x^21 + 59*x^22 - 103*x^23 + 123*x^24 - 54*x^25 + 114*x^26 - 63*x^27 + 62*x^28 - 26*x^29 - 35*x^30 + 11*x^31 - 22*x^32 + 23*x^33 - 28*x^34 + 18*x^35)/(1 - x - 2*x^2 - 7*x^3 - 19*x^5 - 23*x^6 + 85*x^7 + 136*x^8 + 62*x^9 - 72*x^10 + 30*x^11 + 31*x^12 - 36*x^13 - 232*x^14 - 163*x^15 + 78*x^16 - 84*x^17 - 130*x^18 + 12*x^19 + 173*x^20 + 62*x^21 + 93*x^22 - 48*x^23 + 131*x^24 - 134*x^25 + 49*x^26 - 129*x^27 + 58*x^28 - 71*x^29 + 24*x^30 + 38*x^31 - 18*x^32 + 31*x^33 - 23*x^34 + 28*x^35 - 18*x^36).

A332394 Number of minimal total dominating sets in the 4 X n king graph.

Original entry on oeis.org

1, 1, 15, 52, 141, 801, 2950, 9792, 47527, 193530, 748908, 3136113, 12662762, 51628821, 215202537, 877488961, 3564479615, 14617739032, 59930350355, 245634413693, 1006434474838, 4115218924012, 16838191789089, 68996553122238, 282651353770552, 1157272492569645

Offset: 0

Andrew Howroyd, Feb 10 2020

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row n=4 of A332390.

cp's OEIS Frontend

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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A332391 Number of minimal total dominating sets in the n X n king graph.

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A332392 Number of minimal total dominating sets in the 2 X n king graph.

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A332393 Number of minimal total dominating sets in the 3 X n king graph.

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A332394 Number of minimal total dominating sets in the 4 X n king graph.

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