A103134 a(n) = Fibonacci(6n+4).

3, 55, 987, 17711, 317811, 5702887, 102334155, 1836311903, 32951280099, 591286729879, 10610209857723, 190392490709135, 3416454622906707, 61305790721611591, 1100087778366101931, 19740274219868223167, 354224848179261915075, 6356306993006846248183

Offset

0,1

Comments

Gives those numbers which are Fibonacci numbers in A103135.

Generally, for any sequence where a(0)= Fibonacci(p), a(1) = F(p+q) and Lucas(q)*a(1) +- a(0) = F(p+2q), then a(n) = L(q)*a(n-1) +- a(n-2) generates the following Fibonacci sequence: a(n) = F(q(n)+p). So for this sequence, a(n) = 18*a(n-1) - a(n-2) = F(6n+4): q=6, because 18 is the 6th Lucas number (L(0) = 2, L(1)=1); F(4)=3, F(10)=55 and F(16)=987 (F(0)=0 and F(1)=1). See Lucas sequence A000032. This is a special case where a(0) and a(1) are increasing Fibonacci numbers and Lucas(m)*a(1) +- a(0) is another Fibonacci. - Bob Selcoe, Jul 08 2013

a(n) = x + y where x and y are solutions to x^2 = 5*y^2 - 1. (See related sequences with formula below.) - Richard R. Forberg, Sep 05 2013

Links

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

Tanya Khovanova, Recursive Sequences

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

Index entries for sequences related to Chebyshev polynomials.

Formula

G.f.: (x+3)/(x^2-18*x+1).

a(n) = 18*a(n-1) - a(n-2) for n>1; a(0)=3, a(1)=55. - Philippe Deléham, Nov 17 2008

a(n) = A007805(n) + A075796(n), as follows from comment above. - Richard R. Forberg, Sep 05 2013

a(n) = ((15-7*sqrt(5)+(9+4*sqrt(5))^(2*n)*(15+7*sqrt(5))))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016

a(n) = S(3*n+1, 3) = 3*S(n,18) + S(n-1,18), with the Chebyshev S polynomials (A049310), S(-1, x) = 0, and S(n, 18) = A049660(n+1). - Wolfdieter Lang, May 08 2023

Mathematica

Table[Fibonacci[6n+4], {n, 0, 30}]
LinearRecurrence[{18, -1}, {3, 55}, 20] (* Harvey P. Dale, Mar 29 2023 *)
Table[ChebyshevU[3*n+1, 3/2], {n, 0, 20}] (* Vaclav Kotesovec, May 27 2023 *)

Prog

(Magma) [Fibonacci(6*n +4): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011
(PARI) a(n)=fibonacci(6*n+4) \\ Charles R Greathouse IV, Feb 05 2013

Crossrefs

Subsequence of A033887.

Cf. A000032, A000045, A001906, A001519, A015448, A014445, A033888, A033889, A033890, A033891, A049310, A049660, A102312, A099100, A134490, A134491, A134492, A134493, A134494, A134495, A103134, A134497, A134498, A134499, A134500, A134501, A134502, A134503, A134504.

Cf. A103135.

Sequence in context: A197094 A156379 A119192 * A235534 A300992 A304640

Adjacent sequences: A103131 A103132 A103133 * A103135 A103136 A103137

Keyword

nonn,easy

Author

Creighton Dement, Jan 24 2005

Extensions

Edited by N. J. A. Sloane, Aug 10 2010

Status

approved