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A107979
a(n) = 4*a(n-1) + 2*a(n-2) for n>1, with a(0)=2, a(1)=9.
4
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).
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
Sequence in context: A097070 A164033 A020728 * A021001 A231134 A370479
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 12 2005
STATUS
approved