A109113 a(n) = 6*a(n-1) + 3*a(n-2), a(0)=2, a(1)=14.

2, 14, 90, 582, 3762, 24318, 157194, 1016118, 6568290, 42458094, 274453434, 1774094886, 11467929618, 74129862366, 479182963050, 3097487365398, 20022473081538, 129427300585422, 836631222757146, 5408069238299142

Offset: 0

Author

Emeric Deutsch, Jun 19 2005

Keywords

• nonn
• easy

Comments

Kekulé numbers for certain benzenoids.

References

• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 302, P_{15}).

Links

• Index entries for linear recurrences with constant coefficients, signature (6,3).

Crossrefs

Cf. A099842.

Programs

Maple

a[0]:=2: a[1]:=14: for n from 2 to 25 do a[n]:=6*a[n-1]+3*a[n-2] od: seq(a[n],n=0..22);

Mathematica

CoefficientList[Series[2*(1 + x)/(1 - 6*x - 3*x^2), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jan 01 2024 *)

PARI

Vec(2*(1+x)/(1-6*x-3*x^2)+O(x^99)) \\ Charles R Greathouse IV, May 06 2014

Formula

a(n) = ((3 + 2*sqrt(3))^(n+1) + (3 - 2*sqrt(3))^(n+1))/3.

G.f.: 2*(1+z)/(1 - 6*z - 3*z^2).

a(n) = 2*abs(A099842(n)). - F. Chapoton, May 06 2014

E.g.f.: 2*exp(3*x)*(3*cosh(2*sqrt(3)*x) + 2*sqrt(3)*sinh(2*sqrt(3)*x))/3. - Stefano Spezia, Apr 08 2025