A289188 Eternal domination number for P_3 X P_n grid graph.

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

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×n 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