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
Examples
G.f. = 2 + 9*x + 40*x^2 + 178*x^3 + 792*x^4 + 3524*x^5 + 15680*x^6 + 69768*x^7 + ...
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).
Crossrefs
Cf. A021001. - R. J. Mathar, Aug 24 2008
Programs
-
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 *) -
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
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
Comments