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 A056236 #51 Jul 02 2025 16:02:00
%S A056236 2,4,12,40,136,464,1584,5408,18464,63040,215232,734848,2508928,
%T A056236 8566016,29246208,99852800,340918784,1163969536,3974040576,
%U A056236 13568223232,46324811776,158162800640,540001579008,1843680714752,6294719700992
%N A056236 a(n) = (2 + sqrt(2))^n + (2 - sqrt(2))^n.
%C A056236 First differences give A060995. - _Jeremy Gardiner_, Aug 11 2013
%C A056236 Binomial transform of A002203 [Bhadouria].
%C A056236 The binomial transform of this sequence is 2, 6, 22, 90, 386, .. = 2*A083878(n). - _R. J. Mathar_, Nov 10 2013
%H A056236 G. C. Greubel, <a href="/A056236/b056236.txt">Table of n, a(n) for n = 0..1000</a>
%H A056236 Pooja Bhadouria, Deepika Jhala, and Bijendra Singh, <a href="http://dx.doi.org/10.22436/jmcs.08.01.07">Binomial Transforms of the k-Lucas Sequences and its Properties</a>, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92, sequence B_2.
%H A056236 Sela Fried, <a href="https://arxiv.org/abs/2505.14196">Even-up words and their variants</a>, arXiv:2505.14196 [math.CO], 2025. See p. 9.
%H A056236 Takao Komatsu, <a href="https://arxiv.org/abs/2105.08277">Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions</a>, arXiv:2105.08277 [math.CO], 2021.
%H A056236 Youngwoo Kwon, <a href="https://arxiv.org/abs/1804.08119">Binomial transforms of the modified k-Fibonacci-like sequence</a>, arXiv:1804.08119 [math.NT], 2018.
%H A056236 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).
%F A056236 a(n) = 4*a(n-1) - 2*a(n-2).
%F A056236 a(n) = a(n-2) - a(n-1) + 2*A020727(n-1).
%F A056236 a(n) = 2*A006012(n) = 4*A007052(n-1).
%F A056236 For n>2, a(n) = floor((2+sqrt(2))*a(n-1)).
%F A056236 G.f.: 2*(1-2*x)/(1-4*x+2*x^2).
%F A056236 From _L. Edson Jeffery_, Apr 08 2011: (Start)
%F A056236 a(n) = 2^(2*n)*(cos(Pi/8)^(2*n) + cos(3*Pi/8)^(2*n)).
%F A056236 a(n) = 3*a(n-1) + Sum_{k=1..(n-2)} a(k), for n>1, with a(0)=2, a(1)=4. (End)
%F A056236 a(n) = [x^n] ( (1 + 4*x + sqrt(1 + 8*x + 8*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015
%t A056236 LinearRecurrence[{4,-2},{2,4},30] (* _Harvey P. Dale_, Jan 18 2013 *)
%o A056236 (PARI) a(n) = 2*real((2+quadgen(8))^n);
%o A056236 (Sage) [lucas_number2(n,4,2) for n in range(37)] # _Zerinvary Lajos_, Jun 25 2008
%Y A056236 Cf. A006012, A007052, A020727.
%K A056236 nonn,easy
%O A056236 0,1
%A A056236 _Henry Bottomley_, Aug 11 2000
%E A056236 More terms from _James Sellers_, Aug 25 2000