A286514 -id:A286514 - cp's OEIS frontend

A286513 Array read by antidiagonals: T(m,n) is the number of independent sets in the stacked prism graph C_m X P_n.

1, 1, 3, 1, 7, 4, 1, 17, 13, 7, 1, 41, 43, 35, 11, 1, 99, 142, 181, 81, 18, 1, 239, 469, 933, 621, 199, 29, 1, 577, 1549, 4811, 4741, 2309, 477, 47, 1, 1393, 5116, 24807, 36211, 26660, 8303, 1155, 76, 1, 3363, 16897, 127913, 276561, 307983, 143697, 30277, 2785, 123

Offset: 1

Andrew Howroyd, May 10 2017

Equivalently, the number of vertex covers in the stacked prism graph C_m X P_n.

			Table starts:
=============================================================
m\n|   1    2     3      4        5         6           7
---|---------------------------------------------------------
1  |   1    1     1      1        1         1           1 ...
2  |   3    7    17     41       99       239         577 ...
3  |   4   13    43    142      469      1549        5116 ...
4  |   7   35   181    933     4811     24807      127913 ...
5  |  11   81   621   4741    36211    276561     2112241 ...
6  |  18  199  2309  26660   307983   3557711    41097664 ...
7  |  29  477  8303 143697  2488431  43089985   746156517 ...
8  |  47 1155 30277 788453 20546803 535404487 13951571713 ...
...

Liang Kai, Table of n, a(n) for n = 1..704 (terms up to a(435) from Andrew Howroyd)
Kai Liang, Independent Set Enumeration and Estimation of Related Constants of Grid Graphs, arXiv:2507.04007 [math.CO], 2025. See p. 13.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Stacked Prism Graph.
Wikipedia, Independent set.

Rows 3..8 are A003688(n+1), A051926, A181989, A181961, A182014, A182019.
Columns 1..4 are A000032, A051927, A050400, A050401.
Main diagonal is A212270.
Cf. A089934 (P_m X P_n), A027683, A286514.

A286914 Number of dominating sets in the stacked prism graph C_n X P_n.

Original entry on oeis.org

1, 11, 383, 38437, 15179657, 23213149395, 135207176568283, 3008600384479080345, 255856229024918222966821, 83138455533359719800266401039, 103225590675437292878209927396154515, 489730115155632619934278544711183719808829

Offset: 1

Andrew Howroyd, May 15 2017

nonn

Stephan Mertens, Table of n, a(n) for n = 1..26
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
Eric Weisstein's World of Mathematics, Dominating Set
Wikipedia, Dominating set

Main diagonal of A286514.

Cf. A284702, A285880.

A285880 Number of dominating sets in the stacked prism graph C_3 X P_n.

Original entry on oeis.org

7, 51, 383, 2865, 21449, 160579, 1202181, 9000177, 67380199, 504444655, 3776545835, 28273267057, 211668986689, 1584668649559, 11863687582089, 88817989227329, 664939560807015, 4978097605819019, 37268734233506311, 279013924866426465

Offset: 1

Andrew Howroyd, May 15 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Dominating Set
Wikipedia, Dominating set

Row 3 of A286514.

Cf. A284702, A286914.

Empirical: a(n) = 7*a(n-1)+3*a(n-2)+5*a(n-3)-a(n-4)-3*a(n-5)-a(n-6)-a(n-7) for n>7.

Empirical G.f.: x*(7+2*x+5*x^2-4*x^3 -3*x^4-2*x^5-x^6)/(1-7*x-3*x^2-5*x^3 +x^4+3*x^5+x^6+x^7).

A375603 Array read by antidiagonals: T(m,n) = domination number of the stacked prism graph C_m X P_n.

Original entry on oeis.org

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

Offset: 1

Stephan Mertens, Aug 20 2024

			Table starts:
====================================
m\n |   1   2   3    4    5    6 ...
----|-------------------------------
  1 |   1   1   1    2    2    2 ...
  2 |   1   2   2    3    3    4 ...
  3 |   1   2   3    4    4    5 ...
  4 |   2   2   3    4    5    6 ...
  5 |   2   3   4    6    7    8 ...
  6 |   2   4   5    6    8    9 ...
 ...

Stephan Mertens, Table of n, a(n) for n = 1..325
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
Eric Weisstein's World of Mathematics, Domination Number.
Eric Weisstein's World of Mathematics, Stacked Prism Graph.

Main diagonal is A375601.

Cf. A286514, A350823, A375566.

A375566 Array read by antidiagonals: T(m,n) = number of minimum dominating sets in the stacked prism graph C_m X P_n.

Original entry on oeis.org

1, 2, 2, 1, 6, 3, 4, 3, 9, 6, 3, 12, 34, 4, 5, 1, 2, 123, 4, 10, 3, 8, 17, 3, 16, 5, 51, 14, 4, 2, 18, 28, 290, 18, 14, 8, 1, 20, 93, 76, 320, 6, 63, 4, 3, 13, 2, 438, 164, 265, 171, 14, 4, 18, 25, 5, 24, 3, 396, 255, 36, 91, 24, 9, 120, 11, 1, 2, 27, 904, 250, 6, 1526, 60, 2052, 25, 22, 3

Offset: 1

Stephan Mertens, Aug 19 2024

			Table starts:
====================================
m\n |   1   2   3    4    5    6 ...
----+-------------------------------
  1 |   1   2   1    4    3    1 ...
  2 |   2   6   3   12    2   17 ...
  3 |   3   9  34  123    3   18 ...
  4 |   6   4   4   16   28   76 ...
  5 |   5  10   5  290  320  265 ...
 ...

Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
Eric Weisstein's World of Mathematics, Minimum Dominating Set.
Eric Weisstein's World of Mathematics, Stacked Prism Graph.

Main diagonal is A375569.
Rows 1..2 are A347633, A347558.
Column 1 is A347538, column 2 is essentially A347634.
Cf. A286514, A350820, A375603.

cp's OEIS Frontend

A286513 Array read by antidiagonals: T(m,n) is the number of independent sets in the stacked prism graph C_m X P_n.

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A286914 Number of dominating sets in the stacked prism graph C_n X P_n.

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A285880 Number of dominating sets in the stacked prism graph C_3 X P_n.

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A375603 Array read by antidiagonals: T(m,n) = domination number of the stacked prism graph C_m X P_n.

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A375566 Array read by antidiagonals: T(m,n) = number of minimum dominating sets in the stacked prism graph C_m X P_n.

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