Voozh

A087404

a(n) = 4*a(n-1) + 5*a(n-2) for n > 1, with a(0) = 2 and a(1) = 4.

2, 4, 26, 124, 626, 3124, 15626, 78124, 390626, 1953124, 9765626, 48828124, 244140626, 1220703124, 6103515626, 30517578124, 152587890626, 762939453124, 3814697265626, 19073486328124, 95367431640626, 476837158203124, 2384185791015626, 11920928955078124, 59604644775390626

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OFFSET

0,1

COMMENTS

Let F(x) = Product_{n>=0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number 1 + F(-1/5) = 2.24761 97788 60361 46849 ... = 2 + 1/(4 + 1/(26 + 1/(124 + 1/(626 + ...)))). See A111317. - Peter Bala, Dec 26 2012

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..1430

Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.

Index entries for linear recurrences with constant coefficients, signature (4,5).

FORMULA

G.f.: (2 - 4*x)/(1 - 4*x - 5*x^2).

a(n) = 5^n + (-1)^n.

From Elmo R. Oliveira, Aug 23 2024: (Start)

E.g.f.: exp(-x)*(exp(6*x) + 1).

a(n) = 2*A081340(n). (End)

MATHEMATICA

CoefficientList[Series[(2 - 4x)/(1 - 4x - 5x^2), {x, 0, 25}], x]

LinearRecurrence[{4, 5}, {2, 4}, 30] (* Harvey P. Dale, May 13 2022 *)

PROG

(SageMath) [lucas_number2(n, 4, -5) for n in range(0, 22)] # Zerinvary Lajos, May 14 2009