A081554 a(n) = sqrt(2)*( (3+2*sqrt(2))^n - (3-2*sqrt(2))^n ).

0, 8, 48, 280, 1632, 9512, 55440, 323128, 1883328, 10976840, 63977712, 372889432, 2173358880, 12667263848, 73830224208, 430314081400, 2508054264192, 14618011503752, 85200014758320, 496582077046168, 2894292447518688

Offset

0,2

Comments

Numbers m such that ceiling( sqrt(m*m/2) )^2 = 4 + m*m/2. - Ctibor O. Zizka, Nov 09 2009

Numbers m such that 2*m^2+16 is a square. - Bruno Berselli, Dec 17 2014

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

Tanya Khovanova, Recursive Sequences

Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.

Index entries for linear recurrences with constant coefficients, signature (6,-1).

Formula

a(n)^2 = 2*A003499(n)^2 - 8.

a(n) = 8*A001109(n).

G.f.: 8*x/(1-6*x+x^2). - Philippe Deléham, Nov 17 2008

a(0)=0, a(1)=8, a(n) = 6*a(n-1) - a(n-2) for n>1. - Philippe Deléham, Sep 19 2009

a(n) = 4*Pell(2*n) = 4*A000129(2*n). - G. C. Greubel, Aug 16 2018

Mathematica

a = 3 + 2Sqrt[2]; b = 3 - 2Sqrt[2]; Table[Simplify[Sqrt[2](a^n - b^n)], {n, 0, 25}]
CoefficientList[Series[8x/(1-6x+x^2), {x, 0, 40}], x]  (* Harvey P. Dale, Mar 11 2011 *)
Table[4 Fibonacci[2 n, 2], {n, 0, 50}] (* G. C. Greubel, Aug 16 2018 *)

Prog

(PARI) x='x+O('x^50); concat([0], Vec(8*x/(1-6*x+x^2))) \\ G. C. Greubel, Aug 16 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(8*x/(1-6*x+x^2))); // G. C. Greubel, Aug 16 2018

Crossrefs

Sequence in context: A371620 A295047 A295375 * A231109 A079743 A079765

Adjacent sequences: A081551 A081552 A081553 * A081555 A081556 A081557

Keyword

nonn,easy

Author

Mario Catalani (mario.catalani(AT)unito.it), Mar 21 2003

Status

approved