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
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) = 1/12*((sqrt(6)-3)(-(2-sqrt(6))^n) + (3+sqrt(6))(2+sqrt(6))^n). - Harvey P. Dale, Jun 21 2011
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
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 12 2005
STATUS
approved