A004295 Expansion of (1+2*x+x^2)/(1-42*x+x^2).

1, 44, 1848, 77572, 3256176, 136681820, 5737380264, 240833289268, 10109260768992, 424348119008396, 17812511737583640, 747701144859504484, 31385635572361604688, 1317448992894327892412, 55301472065989409876616, 2321344377778660886925460

Offset

0,2

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.

Index entries for linear recurrences with constant coefficients, signature (42,-1).

Formula

From Colin Barker, Apr 16 2016: (Start)

a(n) = sqrt(11/10)*((21+2*sqrt(110))^(-n)*(-1+(21+2*sqrt(110))^(2*n))) for n>0.

a(n) = 42*a(n-1) - a(n-2) for n>2.

(End)

a(n) = -(-1)^(2^n)/2 + sqrt(22/5)*sinh(n*log(21+2*sqrt(110))) + 1/2. - Ilya Gutkovskiy, Apr 16 2016

Mathematica

CoefficientList[Series[(1+2*x+x^2)/(1-42*x+x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 25 2012 *)
LinearRecurrence[{42, -1}, {1, 44, 1848}, 20] (* Harvey P. Dale, Oct 21 2024 *)

Prog

(PARI) Vec((1+2*x+x^2)/(1-42*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Crossrefs

Sequence in context: A231262 A223049 A278793 * A162882 A163230 A163748

Adjacent sequences: A004292 A004293 A004294 * A004296 A004297 A004298

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved