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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 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

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Last modified October 21 04:26 EDT 2019. Contains 328291 sequences. (Running on oeis4.)