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

0,2

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).

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

Mathematica

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

Prog

(MuPAD) numlib::fibonacci(6*n) $ n = 0..25; // Zerinvary Lajos, May 09 2008
(Sage) [fibonacci(6*n) for n in range(0, 17)] # Zerinvary Lajos, May 15 2009]
(Magma) [Fibonacci(6*n): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011
(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

Crossrefs

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

Sequence in context: A212703 A065409 A061899 * A067421 A124059 A275867

Adjacent sequences: A134489 A134490 A134491 * A134493 A134494 A134495

Keyword

nonn,easy

Author

Artur Jasinski, Oct 28 2007

Extensions

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

Status

approved