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%I #47 Sep 08 2022 08:44:58
%S 0,1,76,5777,439128,33379505,2537281508,192866774113,14660412114096,
%T 1114384187445409,84707858657965180,6438911642192799089,
%U 489441992665310695944,37204030354205805690833
%N a(n) = Fibonacci(9*n)/34.
%H G. C. Greubel, <a href="/A049669/b049669.txt">Table of n, a(n) for n = 0..530</a>
%H S. Falcon, <a href="http://dx.doi.org/10.5539/jmr.v4n2p97">Generalized Fibonacci Sequences Generated from a k-Fibonacci Sequence</a>, Journal of Mathematics Research Vol. 4, No. 2; April 2012. - From _N. J. A. Sloane_, Sep 22 2012
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Shaoxiong Yuan, <a href="https://arxiv.org/abs/1907.12459">Generalized Identities of Certain Continued Fractions</a>, arXiv:1907.12459 [math.NT], 2019.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (76, 1).
%F G.f.: x/(1-76*x-x^2), 76=L(9)=A000032(9) (Lucas).
%F a(n) = 76*a(n-1) + a(n-2), n>1, a(0)=0, a(1)=1. - _Philippe Deléham_, Nov 23 2008
%F 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
%F 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
%F E.g.f.: exp(38*x)*sinh(17*sqrt(5)*x)/(17*sqrt(5)). - _Stefano Spezia_, Aug 05 2019
%p with (combinat):seq(fibonacci(3*n,4)/17, n=0..13); # _Zerinvary Lajos_, Apr 20 2008
%t Fibonacci[9Range[0,20]]/34 (* or *) LinearRecurrence[{76,1},{0,1},20] (* _Harvey P. Dale_, Jan 20 2013 *)
%o (MuPAD) numlib::fibonacci(9*n)/34 $ n = 0..25; // _Zerinvary Lajos_, May 09 2008
%o (PARI) for(n=0,30, print1(fibonacci(9*n)/34, ", ")) \\ _G. C. Greubel_, Dec 02 2017
%o (Magma) [Fibonacci(9*n)/(34): n in [0..30]]; // _G. C. Greubel_, Dec 02 2017
%Y A column of array A028412.
%Y Cf. A000045.
%K nonn
%O 0,3
%A _Clark Kimberling_
%E More terms from _James A. Sellers_, Jan 20 2000
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