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%I #77 Dec 08 2019 12:07:59
%S 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,
%T 23,24,25,26,27,27,28,29,30,31,31,32,33,34,35,35,36,37,38,39,39,40,41,
%U 42,43,43,44,45,46,47,47,48,49,50,51,51
%N Eternal domination number for P_3 X P_n grid graph.
%C The all-guards move model of eternal domination was introduced by Goddard et al., where it was called the eternal m-security number.
%H Colin Barker, <a href="/A289188/b289188.txt">Table of n, a(n) for n = 1..1000</a>
%H S. Finbow, M. E. Messinger, and M. F. van Bommel, <a href="https://ajc.maths.uq.edu.au/pdf/61/ajc_v61_p156.pdf">Eternal domination on 3×n grid graphs</a>, Australasian Journal of Combinatorics, 61(2) (2015), 156-174. (next 11 terms)
%H S. Finbow and M. F. van Bommel, <a href="https://ajc.maths.uq.edu.au/pdf/76/ajc_v76_p001.pdf">The eternal domination number for 3xn grid graphs</a>, Australasian Journal of Combinatorics, 76(1) (2020), 1-23. (proof of a(26) = 23 and proof of formula)
%H W. Goddard, S. M. Hedetniemi, and S. T. Hedetniemi, <a href="https://www.researchgate.net/publication/249799004_Eternal_Security_in_Graphs">Eternal security in graphs</a>, Journal of Combinatorial Mathematics and Combinatorial Computing, 52 (2005), 169-180.
%H J. L. Goldwasser, W. F. Klostermeyer, and C. M. Mynhardt, <a href="https://www.researchgate.net/publication/268637806_Eternal_Protection_in_Grid_Graphs">Eternal protection in grid graphs</a>, Utilitas Mathematica, 91 (2013), 47-64. (first 14 terms)
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eternal_dominating_set">Eternal dominating set</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F a(n) = a(n-5) + 4 for n > 26.
%F 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
%o (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
%K nonn,easy
%O 1,1
%A _Martin F. van Bommel_, Sep 12 2017
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