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A056236 a(n) = (2+sqrt(2))^n + (2-sqrt(2))^n. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

Table of n, a(n) for n=0..24.

P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the k-Lucas Sequences and its [sic] Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92, sequence B_2.

Index entries for linear recurrences with constant coefficients, signature (4,-2).

FORMULA

a(n) = 4*a(n-1)-2*a(n-2) = a(n-2)-a(n-1)+2*A020727(n-1) = 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).

a(n) = 2^(2*n)*[cos(Pi/8)^(2*n)+cos(3*Pi/8)^2*n]; also, a(n) = 3*a(n-1)+Sum{k=1..n-2} a(k), for n>1, with a(0)=2, a(1)=4. - L. Edson Jeffery, Apr 08 2011

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Sequence in context: A126946 A113179 A214761 * A028329 A204678 A025227

Adjacent sequences:  A056233 A056234 A056235 * A056237 A056238 A056239

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Aug 11 2000

EXTENSIONS

More terms from James A. Sellers, Aug 25 2000

STATUS

approved

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Last modified December 8 17:11 EST 2016. Contains 278946 sequences.