A134492 a(n) = Fibonacci(6*n).

0, 8, 144, 2584, 46368, 832040, 14930352, 267914296, 4807526976, 86267571272, 1548008755920, 27777890035288, 498454011879264, 8944394323791464, 160500643816367088, 2880067194370816120, 51680708854858323072, 927372692193078999176, 16641027750620563662096

Offset: 0

Author

Artur Jasinski, Oct 28 2007

Keywords

• nonn
• easy

Comments

All terms are divisible by 8. - Alonso del Arte, Jul 27 2013

Conjecture: For n >= 2, the terms of this sequence are exactly those Fibonacci numbers which are the sum of the three numbers of a Pythagorean triple (checked up to F(80)). - Felix Huber, Nov 03 2023

Links

• Colin Barker, Table of n, a(n) for n = 0..500
• Hacène Belbachir, Soumeya Merwa Tebtoub and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• Index entries for linear recurrences with constant coefficients, signature (18,-1).

Crossrefs

Cf. A000032, A000045, A008588, A049660, A079343, A014445, A014448, A134493, A134494, A134495, A103134, A134497, A134498.

Programs

Magma

[Fibonacci(6*n): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Mathematica

Table[Fibonacci[6n], {n, 0, 30}]
LinearRecurrence[{18,-1},{0,8},30] (* Harvey P. Dale, Aug 15 2017 *)

MuPAD

numlib::fibonacci(6*n) $ n = 0..25; // Zerinvary Lajos, May 09 2008

PARI

a(n)=fibonacci(6*n) \\ Charles R Greathouse IV, Sep 16 2015

PARI

concat(0, Vec(8*x/(1-18*x+x^2) + O(x^20))) \\ Colin Barker, Jan 24 2016

Sage

[fibonacci(6*n) for n in range(0, 17)] # Zerinvary Lajos, May 15 2009

Formula

a(n) = 18*a(n-1) - a(n-2) = 8*A049660(n). G.f.: 8*x/(1-18*x+x^2). - R. J. Mathar, Feb 16 2010

a(n) = A000045(A008588(n)). - Michel Marcus, Nov 08 2013

a(n) = ((-1+(9+4*sqrt(5))^(2*n)))/(sqrt(5)*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016

a(n) = L(2n-1) * F(2n+1)^2 + L(2n+1) * F(2n-1)^2, where F(n) = A000045(n) and L(n) = A000032(n). - Diego Rattaggi, Nov 12 2020

a(n) = Fibonacci(3*n) * Lucas(3*n) = A000045(3*n) * A000032(3*n) = A014445(n) * A014448(n). - Amiram Eldar, Jan 11 2022

Extensions

Offset corrected by R. J. Mathar, Feb 16 2010