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a(n) = (2 + sqrt(2))^n + (2 - sqrt(2))^n.
10

%I #47 Aug 27 2021 02:30:10

%S 2,4,12,40,136,464,1584,5408,18464,63040,215232,734848,2508928,

%T 8566016,29246208,99852800,340918784,1163969536,3974040576,

%U 13568223232,46324811776,158162800640,540001579008,1843680714752,6294719700992

%N a(n) = (2 + sqrt(2))^n + (2 - sqrt(2))^n.

%C First differences give A060995. - _Jeremy Gardiner_, Aug 11 2013

%C Binomial transform of A002203 [Bhadouria].

%C The binomial transform of this sequence is 2, 6, 22, 90, 386, .. = 2*A083878(n). - _R. J. Mathar_, Nov 10 2013

%H G. C. Greubel, <a href="/A056236/b056236.txt">Table of n, a(n) for n = 0..1000</a>

%H P. Bhadouria, D. Jhala, and B. 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 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 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).

%F a(n) = 4*a(n-1) - 2*a(n-2).

%F a(n) = a(n-2) - a(n-1) + 2*A020727(n-1).

%F a(n) = 2*A006012(n) = 4*A007052(n-1).

%F For n>2, a(n) = floor((2+sqrt(2))*a(n-1)).

%F G.f.: 2*(1-2*x)/(1-4*x+2*x^2).

%F From _L. Edson Jeffery_, Apr 08 2011: (Start)

%F a(n) = 2^(2*n)*(cos(Pi/8)^(2*n) + cos(3*Pi/8)^(2*n)).

%F 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 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 LinearRecurrence[{4,-2},{2,4},30] (* _Harvey P. Dale_, Jan 18 2013 *)

%o (PARI) a(n) = 2*real((2+quadgen(8))^n);

%o (Sage) [lucas_number2(n,4,2) for n in range(37)] # _Zerinvary Lajos_, Jun 25 2008

%Y Cf. A006012, A007052, A020727.

%K nonn,easy

%O 0,1

%A _Henry Bottomley_, Aug 11 2000

%E More terms from _James A. Sellers_, Aug 25 2000