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

Author

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

Keywords

• nonn
• easy

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).

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(8*x/(1-6*x+x^2))); // 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 *)

PARI

x='x+O('x^50); concat([0], Vec(8*x/(1-6*x+x^2))) \\ G. C. Greubel, Aug 16 2018

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