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Kekulé numbers for certain benzenoids.
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%I #19 Jul 22 2022 09:08:22

%S 5,105,2331,52137,1167291,26137809,585284211,13105857753,293470347915,

%T 6571477451169,147150526244067,3295039441438377,73783527649203195,

%U 1652183243687500785,36996190853123920851,828429983702209723257

%N Kekulé numbers for certain benzenoids.

%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 205).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (27,-108,108).

%F a(n) = (3^(n+1) + (12 + 6*sqrt(3))^(n+1)/4 + (12 - 6*sqrt(3))^(n+1)/4)/27.

%F From _Colin Barker_, Aug 30 2013: (Start)

%F a(n) = 27*a(n-1) - 108*a(n-2) + 108*a(n-3).

%F G.f.: -x*(36*x^2 - 30*x + 5) / ((3*x-1)*(36*x^2 - 24*x + 1)). (End)

%F 9*a(n) = 3^n+ 6^n*A001075(n+1). - _R. J. Mathar_, Jul 22 2022

%p a:=n->(3^(n+1)+(12+6*sqrt(3))^(n+1)/4+(12-6*sqrt(3))^(n+1)/4)/27: seq(expand(a(n)),n=1..18);

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, Nov 30 2005