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A103134 a(n) = Fibonacci(6n+4). 12

%I

%S 3,55,987,17711,317811,5702887,102334155,1836311903,32951280099,

%T 591286729879,10610209857723,190392490709135,3416454622906707,

%U 61305790721611591,1100087778366101931,19740274219868223167,354224848179261915075,6356306993006846248183

%N a(n) = Fibonacci(6n+4).

%C Gives those numbers which are Fibonacci numbers in A103135.

%C 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

%C 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

%H Colin Barker, <a href="/A103134/b103134.txt">Table of n, a(n) for n = 0..750</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-1).

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

%F a(n) = 18*a(n-1) - a(n-2) for n>1; a(0)=3, a(1)=55. - _Philippe Deléham_, Nov 17 2008

%F a(n) = A007805(n) + A075796(n), as follows from comment above. - _Richard R. Forberg_, Sep 05 2013

%F 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

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

%o (MAGMA) [Fibonacci(6*n +4): n in [0..100]]; // _Vincenzo Librandi_, Apr 17 2011

%o (PARI) a(n)=fibonacci(6*n+4) \\ _Charles R Greathouse IV_, Feb 05 2013

%Y Subsequence of A033887.

%Y 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.

%Y Cf. A103135.

%K nonn,easy

%O 0,1

%A _Creighton Dement_, Jan 24 2005

%E Edited by _N. J. A. Sloane_, Aug 10 2010

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Last modified November 19 11:09 EST 2019. Contains 329319 sequences. (Running on oeis4.)