A056236 a(n) = (2 + sqrt(2))^n + (2 - sqrt(2))^n.

2, 4, 12, 40, 136, 464, 1584, 5408, 18464, 63040, 215232, 734848, 2508928, 8566016, 29246208, 99852800, 340918784, 1163969536, 3974040576, 13568223232, 46324811776, 158162800640, 540001579008, 1843680714752, 6294719700992

Offset: 0

Author

Henry Bottomley, Aug 11 2000

Keywords

• nonn
• easy

Comments

First differences give A060995. - Jeremy Gardiner, Aug 11 2013

Binomial transform of A002203 [Bhadouria].

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Pooja Bhadouria, Deepika Jhala, and Bijendra Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92, sequence B_2.
• Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 9.
• Takao Komatsu, Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions, arXiv:2105.08277 [math.CO], 2021.
• Youngwoo Kwon, Binomial transforms of the modified k-Fibonacci-like sequence, arXiv:1804.08119 [math.NT], 2018.
• Index entries for linear recurrences with constant coefficients, signature (4,-2).

Crossrefs

Cf. A006012, A007052, A020727.

Programs

Mathematica

LinearRecurrence[{4,-2},{2,4},30] (* Harvey P. Dale, Jan 18 2013 *)

PARI

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

Sage

[lucas_number2(n,4,2) for n in range(37)] # Zerinvary Lajos, Jun 25 2008

Formula

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

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

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

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

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

From L. Edson Jeffery, Apr 08 2011: (Start)

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

a(n) = 3*a(n-1) + Sum_{k=1..(n-2)} a(k), for n>1, with a(0)=2, a(1)=4. (End)

a(n) = [x^n] ( (1 + 4*x + sqrt(1 + 8*x + 8*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

Extensions

More terms from James Sellers, Aug 25 2000