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 A049668 #44 Sep 08 2022 08:44:58
%S A049668 0,1,47,2208,103729,4873055,228929856,10754830177,505248088463,
%T A049668 23735905327584,1115082302307985,52385132303147711,
%U A049668 2460986135945634432,115613963257141670593,5431395286949712883439,255159964523379363851040,11987086937311880388115441
%N A049668 a(n) = Fibonacci(8*n)/21.
%C A049668 This is the Lucas sequence U(47,1). Also partial sums of A333718. This sequence contains all nonnegative integers a(n) such that 2205*a(n)^2 + 4 = b(n)^2 = 2205*a(n-1)*a(n+1) + 2209, where b(n) = a(n+1) - a(n-1) = A087265(n). - _Klaus Purath_, Aug 14 2021
%H A049668 Colin Barker, <a href="/A049668/b049668.txt">Table of n, a(n) for n = 0..500</a>
%H A049668 R. Flórez, R. A. Higuita and A. Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
%H A049668 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A049668 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (47,-1).
%F A049668 G.f.: x/(1-47*x+x^2), 47=L(8)=A000032(8) (Lucas).
%F A049668 a(n) = 47*a(n-1)-a(n-2) ; a(0)=0, a(1)=1. - _Philippe Deléham_, Nov 18 2008
%F A049668 From _Peter Bala_, Apr 03 2015: (Start)
%F A049668 For integer k, 1 + k*(14 - k)*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + k/3*Sum_{n >= 1} Fibonacci(4*n)*x^n )*( 1 + k/3*Sum_{n >= 1} Fibonacci(4*n)*(-x)^n ).
%F A049668 1 + 45*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + Sum_{n >= 1} Lucas(4*n)*x^n )*( 1 + Sum_{n >= 1} Lucas(4*n)*(-x)^n ).
%F A049668 1 - 36*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + 2*Sum_{n >= 1} Fibonacci(4*n+2)*x^n )*( 1 + 2*Sum_{n >= 1} Fibonacci(4*n+2)*(-x)^n ). (End)
%F A049668 a(n) = ((47 + 21*sqrt(5))^(1-n)*(-2^n + (2207 + 987*sqrt(5))^n )) /(2205 + 987*sqrt(5)). - _Colin Barker_, Jun 03 2016
%F A049668 a(n) = (a(n-1)*a(n-2) - 47)/a(n-3), n > 3; a(n) = (a(n-1)^2 - 1)/a(n-2), n > 2. - _Klaus Purath_, Aug 14 2021
%t A049668 Table[Fibonacci[8*n]/21, {n, 15}] (* _Michael De Vlieger_, Apr 03 2015 *)
%o A049668 (MuPAD) numlib::fibonacci(8*n)/21 $ n = 0..25; // _Zerinvary Lajos_, May 09 2008
%o A049668 (PARI) concat(0, Vec(x/(1-47*x+x^2) + O(x^20))) \\ _Colin Barker_, Jun 03 2016
%o A049668 (PARI) for(n=0,30, print1(fibonacci(8*n)/21, ", ")) \\ _G. C. Greubel_, Dec 02 2017
%o A049668 (Magma) [Fibonacci(8*n)/21: n in [0..30]]; // _G. C. Greubel_, Dec 02 2017
%Y A049668 A column of array A028412.
%Y A049668 Cf. A000045.
%K A049668 nonn,easy
%O A049668 0,3
%A A049668 _Clark Kimberling_