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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Jan 20 2000
STATUS
approved