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Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).
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%I #21 Jul 26 2019 06:25:22

%S 0,3,12,54,240,1068,4752,21144,94080,418608,1862592,8287584,36875520,

%T 164077248,730060032,3248394624,14453698560,64311583488,286153731072,

%U 1273238091264,5665259827200,25207515491328,112160581619712,499057357461504,2220550593085440,9880317087264768,43962369535229952

%N Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).

%H S. J. Cyvin and I. Gutman, <a href="https://doi.org/10.1007/978-3-662-00892-8_6">Kekulé structures in benzenoid hydrocarbons</a>, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78, Binet formula page 77).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,2).

%F Conjectured g.f.: 1/(1 - Q(0)) - 1, where Q(k)= 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1) ))))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 13 2013

%F G.f.: -3*x / (2*x^2+4*x-1). a(n)=3*A090017(n). - _Colin Barker_, Aug 29 2013

%p A123348 := proc(n)

%p 3*((2+sqrt(6))^n-(2-sqrt(6))^n)/2/sqrt(6) ;

%p expand(%) ;

%p simplify(%) ;

%p end proc:

%p seq( A123348(n),n=0..30) ; # _R. J. Mathar_, Jul 26 2019

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Oct 10 2006