A004298 Expansion of (1+2*x+x^2)/(1-66*x+x^2).

1, 68, 4488, 296140, 19540752, 1289393492, 85080429720, 5614018968028, 370440171460128, 24443437297400420, 1612896421456967592, 106426720378862460652, 7022550648583465435440, 463381916086129856278388, 30576183911035987048938168, 2017564756212289015373640700

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..500
• 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 (66,-1).

Crossrefs

Pairwise sums of A078989.

Cf. A097316.

Programs

Mathematica

CoefficientList[Series[(1+2*x+x^2)/(1-66*x+x^2),{x,0,50}],x] (* Vincenzo Librandi, Feb 25 2012 *)
LinearRecurrence[{66,-1},{1,68,4488},20] (* Harvey P. Dale, Sep 23 2020 *)

PARI

Vec((1+2*x+x^2)/(1-66*x+x^2) + O(x^50)) \\ Colin Barker, Apr 16 2016

Formula

For n > 0, a(n) = 68*A097316(n-1). - Gerald McGarvey, Jun 16 2007

From Colin Barker, Apr 16 2016: (Start)

a(n) = (sqrt(17)*(33+8*sqrt(17))^(-n)*(-1+(33+8*sqrt(17))^(2*n)))/4 for n>0.

a(n) = 66*a(n-1) - a(n-2) for n>2.

(End)

a(n) = (-(-1)^(2^n) + sqrt(17)*sinh(n*log(33+8*sqrt(17))) + 1)/2. - Ilya Gutkovskiy, Apr 16 2016