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

0,2

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).

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

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 *)

Prog

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

Crossrefs

Pairwise sums of A049629.

Sequence in context: A358365 A089350 A323961 * A053508 A060918 A115100

Adjacent sequences: A004289 A004290 A004291 * A004293 A004294 A004295

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved