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 A106568 #55 Jun 06 2025 08:06:08
%S A106568 0,4,16,80,384,1856,8960,43264,208896,1008640,4870144,23515136,
%T A106568 113541120,548225024,2647064576,12781158400,61712891904,297976201216,
%U A106568 1438756372480,6946930294784,33542746669056,161958707855360,782005818097664,3775858103812096,18231455687639040
%N A106568 Expansion of 4*x/(1 - 4*x - 4*x^2).
%C A106568 This sequence is part of a class of sequences with the properties: a(n) = m*(a(n-1) + a(n-2)) with a(0) = 0 and a(1) = m, g.f.: m*x/(1 - m*x - m*x^2), and have the Binet form m*(alpha^n - beta^n)/(alpha - beta) where 2*alpha = m + sqrt(m^2 + 4*m) and 2*beta = p - sqrt(m^2 + 4*m). - _G. C. Greubel_, Sep 06 2021
%H A106568 G. C. Greubel, <a href="/A106568/b106568.txt">Table of n, a(n) for n = 0..1000</a>
%H A106568 Martin Burtscher, Igor Szczyrba and Rafał Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H A106568 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A106568 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,4).
%F A106568 a(n) = 4 * A057087(n).
%F A106568 a(n) = A094013(n+1). - _R. J. Mathar_, Aug 24 2008
%F A106568 From _Philippe Deléham_, Sep 19 2009: (Start)
%F A106568 a(n) = 4*a(n-1) + 4*a(n-2) for n > 2; a(0) = 0, a(1)=4.
%F A106568 G.f.: 4*x/(1 - 4*x - 4*x^2). (End)
%F A106568 G.f.: Q(0) - 1, where Q(k) = 1 + 2*(1+2*x)*x + 2*(2*k+3)*x - 2*x*(2*k+1 +2*x+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 04 2013
%F A106568 a(n) = 2^(n+1)*A000129(n). - _G. C. Greubel_, Sep 06 2021
%F A106568 a(n) = 4^n*hypergeom([(1-n)/2, 1-n/2], [1-n], -1) for n > 0. - _Peter Luschny_, Mar 30 2025
%p A106568 A106568 := n -> ifelse(n=0, 0, 4^(n)*hypergeom([(1-n)/2, 1-n/2], [1-n], -1)):
%p A106568 seq(simplify(A106568(n)), n = 0..24); # _Peter Luschny_, Mar 30 2025
%t A106568 LinearRecurrence[{4,4}, {0,4}, 40] (* _G. C. Greubel_, Sep 06 2021 *)
%o A106568 (Magma) [n le 2 select 4*(n-1) else 4*(Self(n-1) +Self(n-2)): n in [1..41]]; // _G. C. Greubel_, Sep 06 2021
%o A106568 (Sage) [2^(n+1)*lucas_number1(n,2,-1) for n in (0..40)] # _G. C. Greubel_, Sep 06 2021
%Y A106568 Cf. A000129, A057087, A009013.
%Y A106568 Sequences of the form a(n) = m*(a(n-1) + a(n-2)): A000045 (m=1), A028860 (m=2), A106435 (m=3), A094013 (m=4), A106565 (m=5), A221461 (m=6), A221462 (m=7).
%K A106568 nonn,easy,less
%O A106568 0,2
%A A106568 _Roger L. Bagula_, May 30 2005
%E A106568 Edited by _N. J. A. Sloane_, Apr 30 2006
%E A106568 Simpler name using o.g.f. by _Joerg Arndt_, Oct 05 2013