A107979 a(n) = 4*a(n-1) + 2*a(n-2) for n>1, with a(0)=2, a(1)=9.

2, 9, 40, 178, 792, 3524, 15680, 69768, 310432, 1381264, 6145920, 27346208, 121676672, 541399104, 2408949760, 10718597248, 47692288512, 212206348544, 944209971200, 4201252581888, 18693430269952, 83176226243584

Offset

0,1

Comments

Kekulé numbers for certain benzenoids.

This is the case r=2 of the generalized Pell numbers as defined in Bród. - Michel Marcus, Oct 28 2020

References

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 78).

Links

Table of n, a(n) for n=0..21.

Dorota Bród, On a New One Parameter Generalization of Pell Numbers, Annales Mathematicae Silesianae 33 (2019), 66-76.

Index entries for linear recurrences with constant coefficients, signature (4,2).

Formula

From R. J. Mathar, Aug 24 2008: (Start)

O.g.f.: (2+x)/(1-4x-2x^2).

a(n) = 2*A090017(n) + A090017(n-1). (End)

a(n) = 1/12*((sqrt(6)-3)(-(2-sqrt(6))^n) + (3+sqrt(6))(2+sqrt(6))^n). - Harvey P. Dale, Jun 21 2011

a(n) = A000129(n+2) + Sum_{k=1..n} A000129(k+1)*a(n-k). - Ralf Stephan, May 23 2014

Example

G.f. = 2 + 9*x + 40*x^2 + 178*x^3 + 792*x^4 + 3524*x^5 + 15680*x^6 + 69768*x^7 + ...

Maple

a[0]:=2: a[1]:=9: for n from 2 to 26 do a[n]:=4*a[n-1]+2*a[n-2] od: seq(a[n], n=0..26);

Mathematica

LinearRecurrence[{4, 2}, {2, 9}, 30] (* or *) CoefficientList[Series[(-x-2)/(2x^2+4x-1), {x, 0, 30}], x] (* Harvey P. Dale, Jun 21 2011 *)
a[ n_] := With[{m = n + 2}, If[ m < 0, -(-2)^m, 1] SeriesCoefficient[ x / (2 - 8 x - 4 x^2), {x, 0, Abs@m}]]; (* Michael Somos, May 23 2014 *)
a[ n_] := With[{m = n + 2, r = Sqrt[6]}, If[ m < 0, -(-2)^m, Sign@m] Expand[(2 + r)^(Abs@m) / (2 r)][[1]]]; (* Michael Somos, May 23 2014 *)

Prog

(PARI) {a(n) = my(m = n+2); if( m<0, -(-2)^m, 1) * polcoeff( x / (2 - 8*x - 4*x^2) + x * O(x^abs(m)), abs(m))}; /* Michael Somos, May 23 2014 */
(PARI) {a(n) = my(r = 2 + quadgen(24)); imag( (1 + 2*r) * r^n)}; /* Michael Somos, May 23 2014 */
(PARI) a(n)=([0, 1; 2, 4]^n*[2; 9])[1, 1] \\ Charles R Greathouse IV, Feb 07 2017

Crossrefs

Cf. A021001. - R. J. Mathar, Aug 24 2008

Cf. A000129, A090017.

Sequence in context: A097070 A164033 A020728 * A021001 A231134 A370479

Adjacent sequences: A107976 A107977 A107978 * A107980 A107981 A107982

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 12 2005

Status

approved