A134497 a(n) = Fibonacci(6n+5).

5, 89, 1597, 28657, 514229, 9227465, 165580141, 2971215073, 53316291173, 956722026041, 17167680177565, 308061521170129, 5527939700884757, 99194853094755497, 1779979416004714189, 31940434634990099905, 573147844013817084101, 10284720757613717413913

Offset

0,1

Links

Colin Barker, Table of n, a(n) for n = 0..750

Index entries for linear recurrences with constant coefficients, signature (18,-1).

Index entries for sequences related to Chebyshev polynomials.

Formula

G.f.: ( 5-x ) / ( 1-18*x+x^2 ). a(n) = 5*A049660(n+1)-A049660(n). - R. J. Mathar, Apr 17 2011

a(n) = A000045(A016969(n)). - Michel Marcus, Nov 08 2013

a(n) = ((25-11*sqrt(5)+(9+4*sqrt(5))^(2*n)*(25+11*sqrt(5))))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016

a(n) = 5*S(n, 18) - S(n-1, 18), n >= 0, with the Chebyshev S-polynomials S(n-1, 18) = A049660(n). (See the g.f.) - Wolfdieter Lang, Jul 10 2018

From Peter Bala, Aug 11 2022: (Start)

Let n ** m = n*m + floor(phi*n)*floor(phi*m), where phi = (1 + sqrt(5))/2, denote the Porta-Stolarsky star product of the integers n and m (see A101858). Then a(n) = 5 ** 5 ** ... ** 5 (n+1 factors).

a(2*n+1) = a(n) ** a(n) = Fibonacci(12*n+11); a(3*n+2) = a(n) ** a(n) ** a(n) = Fibonacci(18*n+17) and so on. (End)

Mathematica

Table[Fibonacci[6n+5], {n, 0, 30}]
Take[Fibonacci[Range[100]], {5, -1, 6}] (* Harvey P. Dale, Jun 18 2013 *)

Prog

(Magma) [Fibonacci(6*n +5): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011
(PARI) a(n)=fibonacci(6*n+5) \\ Charles R Greathouse IV, Jun 11 2015
(PARI) Vec((5-x)/(1-18*x+x^2) + O(x^100)) \\ Altug Alkan, Jan 24 2016

Crossrefs

Cf. A000045, A049660, A101858, A103134, A134492, A134493, A134494, A134495.

Sequence in context: A348578 A360064 A067257 * A358388 A339001 A330605

Adjacent sequences: A134494 A134495 A134496 * A134498 A134499 A134500

Keyword

nonn,easy

Author

Artur Jasinski, Oct 28 2007

Extensions

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

Status

approved