A004294 Expansion of (1+2*x+x^2)/(1-34*x+x^2).

1, 36, 1224, 41580, 1412496, 47983284, 1630019160, 55372668156, 1881040698144, 63900011068740, 2170719335639016, 73740557400657804, 2505008232286726320, 85096539340348037076, 2890777329339546534264, 98201332658204234127900, 3335954533049604413814336

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• 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..600
• J. M. Alonso, Growth functions of amalgams, in Alperin, ed., Arboreal Group Theory, Springer, pp. 1-34, esp. p. 32.
• 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 (34,-1).

Crossrefs

Pairwise sums of A046176.

Programs

Mathematica

CoefficientList[Series[(1+2*x+x^2)/(1-34*x+x^2),{x,0,20}],x] (* Vincenzo Librandi, Jun 14 2012 *)
LinearRecurrence[{34,-1},{1,36,1224},20] (* Harvey P. Dale, Mar 29 2019 *)

PARI

Vec((1+2*x+x^2)/(1-34*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

From Colin Barker, Apr 16 2016: (Start)

a(n) = 3*((17+12*sqrt(2))^(1-n)*(-1+(17+12*sqrt(2))^(2*n)))/(48+34*sqrt(2)) for n>0.

a(n) = 34*a(n-1) - a(n-2) for n>2.

(End)

a(n) = (-(-1)^(2^n) + 3*sqrt(2)*sinh(n*log(17+12*sqrt(2))) + 1)/2. - Ilya Gutkovskiy, Apr 16 2016