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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

%I #27 Oct 28 2020 03:56:50

%S 2,9,40,178,792,3524,15680,69768,310432,1381264,6145920,27346208,

%T 121676672,541399104,2408949760,10718597248,47692288512,212206348544,

%U 944209971200,4201252581888,18693430269952,83176226243584

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

%C Kekulé numbers for certain benzenoids.

%C This is the case r=2 of the generalized Pell numbers as defined in Bród. - _Michel Marcus_, Oct 28 2020

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

%H Dorota Bród, <a href="https://doi.org/10.2478/amsil-2019-0011">On a New One Parameter Generalization of Pell Numbers</a>, Annales Mathematicae Silesianae 33 (2019), 66-76.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,2).

%F From _R. J. Mathar_, Aug 24 2008: (Start)

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

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

%F a(n) = 1/12*((sqrt(6)-3)(-(2-sqrt(6))^n) + (3+sqrt(6))(2+sqrt(6))^n). - _Harvey P. Dale_, Jun 21 2011

%F a(n) = A000129(n+2) + Sum_{k=1..n} A000129(k+1)*a(n-k). - _Ralf Stephan_, May 23 2014

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

%p 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);

%t 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 *)

%t 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 *)

%t 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 *)

%o (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 */

%o (PARI) {a(n) = my(r = 2 + quadgen(24)); imag( (1 + 2*r) * r^n)}; /* _Michael Somos_, May 23 2014 */

%o (PARI) a(n)=([0,1; 2,4]^n*[2;9])[1,1] \\ _Charles R Greathouse IV_, Feb 07 2017

%Y Cf. A021001. - _R. J. Mathar_, Aug 24 2008

%Y Cf. A000129, A090017.

%K nonn,easy

%O 0,1

%A _Emeric Deutsch_, Jun 12 2005

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)