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Number of states in the evolutionary spatial prisoner's dilemma with n players.
1

%I #8 Aug 21 2021 13:19:27

%S 2,2,2,3,4,6,9,13,19,28,42,63,95,143,216,327,496,754,1147,1747,2662,

%T 4059,6192,9450,14428,22034,33658,51422,78573,120073,183510,280485,

%U 428733,655371,1001854,1531567,2341417,3579571,5472565,8366756

%N Number of states in the evolutionary spatial prisoner's dilemma with n players.

%C Conjecture: satisfies a linear recurrence having signature (2, 0, -2, 1, 2, -2, 0, 0, 0, -1). - _Harvey P. Dale_, Aug 21 2021

%H Burger, A. P. ; Van Der Merwe, M.; Van Vuuren, J. H. <a href="https://doi.org/10.1016/j.dam.2012.04.022">An asymptotic analysis of the evolutionary spatial prisoner’s dilemma on a path</a>, Discrete Appl. Math. 160, No. 15, 2075-2088 (2012) Table 4.2

%F Conjecture: g.f. 2*x -x^2*(-2 +2*x +x^2 -2*x^3 +3*x^5 +x^7 +x^9)/ (x^5+x^2-1)/ (x^5-x^2+2*x-1) .

%p A308620 := proc(n)

%p add( binomial(n-3*i+2,2*i-2) +binomial(floor((n-5*i+4)/2)+i-1,i-1), i=1..floor((n+4)/5)) ;

%p %+add(binomial(n-3*i-2,2*i)+binomial(floor((n-5*i-2)/2)+i,i), i=1..floor((n-2)/5)) ;

%p %/2+1 ;

%p %+add(binomial(n-3*i,2*i-1),i=1..floor((n+1)/5)) ;

%p end proc:

%p seq(A308620(n),n=1..40) ;

%K nonn,easy

%O 1,1

%A _R. J. Mathar_, Jun 11 2019