A004292 Expansion of (1+x)^2/(1-18*x+x^2).

1, 20, 360, 6460, 115920, 2080100, 37325880, 669785740, 12018817440, 215668928180, 3870021889800, 69444725088220, 1246135029698160, 22360985809478660, 401251609540917720, 7200167985927040300

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• J. M. Alonso, Growth functions of amalgams, in Alperin, ed., Arboreal Group Theory, Springer, pp. 1-34, esp. p. 32.
• P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..800
• Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• Index entries for linear recurrences with constant coefficients, signature (18,-1).

Crossrefs

Pairwise sums of A049629.

Programs

Maple

f:= gfun:-rectoproc({a(n)=18*a(n-1)-a(n-2),a(0)=1,a(1)=20,a(2)=360},a(n),remember):
map(f, [$0..20]); # Robert Israel, Jun 01 2015

Mathematica

CoefficientList[Series[(1+x)^2/(1-18*x+x^2),{x,0,20}],x] (* Vincenzo Librandi, Jun 13 2012 *)
a[n_]:=1/2(1-(-1)^2^n+(20+9 Sqrt[5])((9+4 Sqrt[5])^(2 n)-1)/(9+4 Sqrt[5])^(n+1));Table[a[n] // FullSimplify,{n,0,20}] (* Gerry Martens, May 30 2015 *)

PARI

Vec((1+x)^2/(1-18*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (1/2)*(1 - (-1)^2^n + (20+9*sqrt(5))*((9+4*sqrt(5))^(2*n) - 1)/(9+4*sqrt(5))^(n+1)). - Gerry Martens, May 30 2015