OFFSET
0,2
COMMENTS
For positive n, a(n) equals (-1)^n times the permanent of the (6n)X(6n) tridiagonal matrix with ((-1)^(1/6))'s along the three central diagonals. - John M. Campbell, Jul 12 2011
a(n) = x + y where those two values are solutions to: x^2 = 5*y^2 + 1. (See related sequences with formula below). - Richard R. Forberg, Sep 05 2013
LINKS
FORMULA
From R. J. Mathar, Apr 17 2011: (Start)
G.f.: ( 1-5*x ) / ( 1-18*x+x^2 ).
a(n) = Fibonacci(3*n+1)^2 + Fibonacci(3*n)^2. - Gary Detlefs, Oct 12 2011
a(n) = 18*a(n-1) - a(n-2). - Richard R. Forberg, Sep 05 2013
a(n) = ((5-sqrt(5)+(5+sqrt(5))*(9+4*sqrt(5))^(2*n)))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016
2*a(n) = Fibonacci(6*n) + Lucas(6*n). - Bruno Berselli, Oct 13 2017
a(n) = S(n, 18) - 5*S(n-1, 18), n >= 0, with the Chebyshev S-polynomials S(n-1, 18) = A049660(n). (See the g.f.) - Wolfdieter Lang, Jul 10 2018
MATHEMATICA
Table[Fibonacci[6n+1], {n, 0, 30}]
PROG
(Magma) [Fibonacci(6*n+1): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011
(PARI) a(n)=fibonacci(6*n+1) \\ Charles R Greathouse IV, Jul 15 2011
(PARI) Vec((1-5*x)/(1-18*x+x^2) + O(x^100)) \\ Altug Alkan, Jan 24 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Oct 28 2007
EXTENSIONS
Offset changed to 0 by Vincenzo Librandi, Apr 16 2011
STATUS
approved