This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A103134 #48 May 27 2023 03:52:55
%S A103134 3,55,987,17711,317811,5702887,102334155,1836311903,32951280099,
%T A103134 591286729879,10610209857723,190392490709135,3416454622906707,
%U A103134 61305790721611591,1100087778366101931,19740274219868223167,354224848179261915075,6356306993006846248183
%N A103134 a(n) = Fibonacci(6n+4).
%C A103134 Gives those numbers which are Fibonacci numbers in A103135.
%C A103134 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 A103134 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 A103134 Colin Barker, <a href="/A103134/b103134.txt">Table of n, a(n) for n = 0..750</a>
%H A103134 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A103134 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-1).
%H A103134 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A103134 G.f.: (x+3)/(x^2-18*x+1).
%F A103134 a(n) = 18*a(n-1) - a(n-2) for n>1; a(0)=3, a(1)=55. - _Philippe Deléham_, Nov 17 2008
%F A103134 a(n) = A007805(n) + A075796(n), as follows from comment above. - _Richard R. Forberg_, Sep 05 2013
%F A103134 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 A103134 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 A103134 Table[Fibonacci[6n+4], {n, 0, 30}]
%t A103134 LinearRecurrence[{18,-1},{3,55},20] (* _Harvey P. Dale_, Mar 29 2023 *)
%t A103134 Table[ChebyshevU[3*n+1, 3/2], {n, 0, 20}] (* _Vaclav Kotesovec_, May 27 2023 *)
%o A103134 (Magma) [Fibonacci(6*n +4): n in [0..100]]; // _Vincenzo Librandi_, Apr 17 2011
%o A103134 (PARI) a(n)=fibonacci(6*n+4) \\ _Charles R Greathouse IV_, Feb 05 2013
%Y A103134 Subsequence of A033887.
%Y A103134 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 A103134 Cf. A103135.
%K A103134 nonn,easy
%O A103134 0,1
%A A103134 _Creighton Dement_, Jan 24 2005
%E A103134 Edited by _N. J. A. Sloane_, Aug 10 2010