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) = ((-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
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Oct 28 2007
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 16 2010
STATUS
approved