A109113 - OEIS

%I #32 Apr 08 2025 03:34:04

%S 2,14,90,582,3762,24318,157194,1016118,6568290,42458094,274453434,

%T 1774094886,11467929618,74129862366,479182963050,3097487365398,

%U 20022473081538,129427300585422,836631222757146,5408069238299142

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

%C 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. 302, P_{15}).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,3).

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

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

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

%F 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

%p 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);

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

%o (PARI) Vec(2*(1+x)/(1-6*x-3*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, May 06 2014

%Y Cf. A099842.

%K nonn,easy

%O 0,1

%A _Emeric Deutsch_, Jun 19 2005