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The Merrifield-Simmons index of the meta-polyphenylene chain of length n.
2

%I #10 Sep 01 2019 18:29:02

%S 1,18,299,4982,83001,1382818,23038099,383820582,6394548401,

%T 106534800818,1774896845499,29570232336982,492647582692201,

%U 8207633878782018,136741265470126499,2278144220188626182,37954461443189410401,632330969513105454418,10534794588084770785099

%N The Merrifield-Simmons index of the meta-polyphenylene chain of length n.

%C The Merrifield-Simmons index of a graph is the number of its independent vertex subsets.

%D R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989.

%H T. Doslic, M. S. Litz, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match67/n2/match67n2_313-330.pdf">Matchings and independent sets in polyphenylene chains</a>, MATCH, Commun. Math. Comput. Chem., 67, 2012, 313-330.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,11).

%F a(n) = (((2+sqrt(3))*(8+5*sqrt(3))^n - (2-sqrt(3))*(8-5*sqrt(3))^n)*(1/2))/sqrt(3).

%F G.f.: (1+2*x)/(1-16*x -11*x^2).

%p gser := series((2*x+1)/(1-16*x-11*x^2), x = 0, 22): seq(coeff(gser, x, n), n = 0 .. 20);

%Y Cf. A228604, A228606.

%K nonn,easy

%O 0,2

%A _Emeric Deutsch_, Nov 23 2013