A049669 a(n) = Fibonacci(9*n)/34.

0, 1, 76, 5777, 439128, 33379505, 2537281508, 192866774113, 14660412114096, 1114384187445409, 84707858657965180, 6438911642192799089, 489441992665310695944, 37204030354205805690833

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..530

S. Falcon, Generalized Fibonacci Sequences Generated from a k-Fibonacci Sequence, Journal of Mathematics Research Vol. 4, No. 2; April 2012. - From N. J. A. Sloane, Sep 22 2012

Tanya Khovanova, Recursive Sequences

Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.

Index entries for linear recurrences with constant coefficients, signature (76, 1).

Formula

G.f.: x/(1-76*x-x^2), 76=L(9)=A000032(9) (Lucas).

a(n) = 76*a(n-1) + a(n-2), n>1, a(0)=0, a(1)=1. - Philippe Deléham, Nov 23 2008

a(n) = (9*F(n) + (-1)^n*30*5*F(n)^3 + 27*5^2*F(n)^5 + (-1)^n*9*5^3*F(n)^7 + 5^4*F(n)^9)/34, n >= 0. See the general D. Jennings formula given in a comment on the triangle A111125, where also the reference is given. Here the fifth row (k=4) applies. - Wolfdieter Lang, Sep 01 2012

For n >= 1, a(n) equals the denominator of the continued fraction [76, 76, ..., 76] (with n copies of 76). The numerator of that continued fraction is a(n+1). - Greg Dresden and Shaoxiong Yuan, Jul 26 2019

E.g.f.: exp(38*x)*sinh(17*sqrt(5)*x)/(17*sqrt(5)). - Stefano Spezia, Aug 05 2019

Maple

with (combinat):seq(fibonacci(3*n, 4)/17, n=0..13); # Zerinvary Lajos, Apr 20 2008

Mathematica

Fibonacci[9Range[0, 20]]/34 (* or *) LinearRecurrence[{76, 1}, {0, 1}, 20] (* Harvey P. Dale, Jan 20 2013 *)

Prog

(MuPAD) numlib::fibonacci(9*n)/34 $ n = 0..25; // Zerinvary Lajos, May 09 2008
(PARI) for(n=0, 30, print1(fibonacci(9*n)/34, ", ")) \\ G. C. Greubel, Dec 02 2017
(Magma) [Fibonacci(9*n)/(34): n in [0..30]]; // G. C. Greubel, Dec 02 2017

Crossrefs

A column of array A028412.

Cf. A000045.

Sequence in context: A116264 A324434 A004299 * A198476 A234778 A139671

Adjacent sequences: A049666 A049667 A049668 * A049670 A049671 A049672

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from James A. Sellers, Jan 20 2000

Status

approved