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A103134 a(n) = Fibonacci(6n+4). 12
3, 55, 987, 17711, 317811, 5702887, 102334155, 1836311903, 32951280099, 591286729879, 10610209857723, 190392490709135, 3416454622906707, 61305790721611591, 1100087778366101931, 19740274219868223167, 354224848179261915075, 6356306993006846248183 (list; graph; refs; listen; history; text; internal format)
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).

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

MATHEMATICA

Table[Fibonacci[6n+4], {n, 0, 30}]

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, 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 A015099 A002818

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

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Last modified November 24 00:27 EST 2017. Contains 295164 sequences.