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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. 2
 2, 9, 40, 178, 792, 3524, 15680, 69768, 310432, 1381264, 6145920, 27346208, 121676672, 541399104, 2408949760, 10718597248, 47692288512, 212206348544, 944209971200, 4201252581888, 18693430269952, 83176226243584 (list; graph; refs; listen; history; text; internal format)
 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) = 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 A038112 Adjacent sequences:  A107976 A107977 A107978 * A107980 A107981 A107982 KEYWORD nonn,easy AUTHOR Emeric Deutsch, Jun 12 2005 STATUS approved

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Last modified June 15 08:14 EDT 2021. Contains 345048 sequences. (Running on oeis4.)