A123348 Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).

0, 3, 12, 54, 240, 1068, 4752, 21144, 94080, 418608, 1862592, 8287584, 36875520, 164077248, 730060032, 3248394624, 14453698560, 64311583488, 286153731072, 1273238091264, 5665259827200, 25207515491328, 112160581619712, 499057357461504, 2220550593085440, 9880317087264768, 43962369535229952

Offset: 0

Author

N. J. A. Sloane, Oct 10 2006

Keywords

• nonn
• easy

Links

• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78, Binet formula page 77).
• Index entries for linear recurrences with constant coefficients, signature (4,2).

Programs

Maple

A123348 := proc(n)
    3*((2+sqrt(6))^n-(2-sqrt(6))^n)/2/sqrt(6) ;
    expand(%) ;
    simplify(%) ;
end proc:
seq( A123348(n),n=0..30) ; # R. J. Mathar, Jul 26 2019

Formula

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

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