

A289188


Eternal domination number for P_3 X P_n grid graph.


1



2, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 47, 48, 49, 50, 51, 51
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OFFSET

1,1


COMMENTS

The allguards move model of eternal domination was introduced by Goddard et al., where it was called the eternal msecurity number.


LINKS

Colin Barker, Table of n, a(n) for n = 1..1000
S. Finbow, M. E. Messinger, and M. F. van Bommel, Eternal domination on 3×n grid graphs, Australasian Journal of Combinatorics, 61(2) (2015), 156174. (next 11 terms)
S. Finbow and M. F. van Bommel, The eternal domination number for 3xn grid graphs, Australasian Journal of Combinatorics, 76(1) (2020), 123. (proof of a(26) = 23 and proof of formula)
W. Goddard, S. M. Hedetniemi, and S. T. Hedetniemi, Eternal security in graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, 52 (2005), 169180.
J. L. Goldwasser, W. F. Klostermeyer, and C. M. Mynhardt, Eternal protection in grid graphs, Utilitas Mathematica, 91 (2013), 4764. (first 14 terms)
Wikipedia, Eternal dominating set
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,1).


FORMULA

a(n) = a(n5) + 4 for n > 26.
G.f.: x*(2 + x^2 + x^3 + x^4  x^5 + x^6  x^8 + x^13  x^15 + x^25  x^26) / ((1  x)^2*(1 + x + x^2 + x^3 + x^4)).  Colin Barker, Sep 12 2017


PROG

(PARI) Vec(x*(2 + x^2 + x^3 + x^4  x^5 + x^6  x^8 + x^13  x^15 + x^25  x^26) / ((1  x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^100)) \\ Colin Barker, Sep 13 2017


CROSSREFS

Sequence in context: A078171 A157282 A114010 * A111633 A273662 A034138
Adjacent sequences: A289185 A289186 A289187 * A289189 A289190 A289191


KEYWORD

nonn,easy


AUTHOR

Martin F. van Bommel, Sep 12 2017


STATUS

approved



