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

1,1

COMMENTS

The all-guards move model of eternal domination was introduced by Goddard et al., where it was called the eternal m-security 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√ón grid graphs, Australasian Journal of Combinatorics, 61(2) (2015), 156-174. (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), 1-23. (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), 169-180.

J. L. Goldwasser, W. F. Klostermeyer, and C. M. Mynhardt, Eternal protection in grid graphs, Utilitas Mathematica, 91 (2013), 47-64. (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(n-5) + 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

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Last modified September 24 04:19 EDT 2020. Contains 337317 sequences. (Running on oeis4.)