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 A190969 #47 Dec 23 2023 09:43:02
%S A190969 0,1,5,17,45,89,85,-287,-2115,-8279,-24475,-56143,-84915,24569,802165,
%T A190969 3814273,12654045,32756041,62547845,50690897,-246928275,-1640168551,
%U A190969 -6225416555,-18005734367,-40225339395,-57080822039,36398604965,638639601137,2902009165965
%N A190969 a(n) = 5*a(n-1) - 8*a(n-2), with a(0)=0, a(1)=1.
%C A190969 Let S(p):=Sum_{k=0..p-1} a(4k)*binomial(2k,k)^3/(-4096)^k. Zhi-Wei Sun conjectured that S(p) == 0 (mod p^2) for every odd prime p, and also S(p) == 0 (mod p^3) for any odd prime p == 1,2,4 (mod 7). - _Zhi-Wei Sun_, Mar 13 2013
%C A190969 (a(n) + ((-1)^n)*n) mod 7 = 0 for n > 0; division yields following signed integer sequence: {0, 1, 2, 7, 12, 13, -42, -301, -1184, -3495, -8022, -12129, 3508, 114597, ...} with g.f.: (x - x^2)/((1 + x)^2 * (1 - 5*x + 8*x^2)). - _Alexander R. Povolotsky_, Mar 13 2013
%H A190969 Zhi-Wei Sun, <a href="/A190969/b190969.txt">Table of n, a(n) for n = 0..100</a>
%H A190969 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2017.
%H A190969 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8).
%F A190969 G.f.: x/(1-5x+8*x^2). - _Philippe Deléham_, Oct 12 2011
%t A190969 LinearRecurrence[{5,-8}, {0,1}, 50]
%o A190969 (Maxima) a[0]:0$ a[1]:1$ a[n]:=5*a[n-1] - 8*a[n-2]$ makelist(a[n], n,0, 50); /* _Martin Ettl_, Oct 21 2012 */
%Y A190969 Cf. A190958 (index to generalized Fibonacci sequences).
%K A190969 sign,easy
%O A190969 0,3
%A A190969 _Vladimir Joseph Stephan Orlovsky_, May 24 2011