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%I #48 May 27 2023 03:52:55
%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).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%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
%F 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
%t Table[Fibonacci[6n+4], {n, 0, 30}]
%t LinearRecurrence[{18,-1},{3,55},20] (* _Harvey P. Dale_, Mar 29 2023 *)
%t Table[ChebyshevU[3*n+1, 3/2], {n, 0, 20}] (* _Vaclav Kotesovec_, May 27 2023 *)
%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, A049310, A049660, 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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