A134493 a(n) = Fibonacci(6*n+1).

1, 13, 233, 4181, 75025, 1346269, 24157817, 433494437, 7778742049, 139583862445, 2504730781961, 44945570212853, 806515533049393, 14472334024676221, 259695496911122585, 4660046610375530309, 83621143489848422977, 1500520536206896083277, 26925748508234281076009

Offset: 0

Author

Artur Jasinski, Oct 28 2007

Keywords

• nonn
• easy

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

• Colin Barker, Table of n, a(n) for n = 0..750
• Index entries for linear recurrences with constant coefficients, signature (18,-1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A000032, A000045, A049660, A134492, A134494, A134495, A103134, A134497.

Programs

Magma

[Fibonacci(6*n+1): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011

Mathematica

Table[Fibonacci[6n+1], {n, 0, 30}]

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

Formula

From R. J. Mathar, Apr 17 2011: (Start)

G.f.: ( 1-5*x ) / ( 1-18*x+x^2 ).

a(n) = A049660(n+1) - 5*A049660(n). (End)

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) = A060645(n) + A023039(n), as derives from comment above. - 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

Extensions

Offset changed to 0 by Vincenzo Librandi, Apr 16 2011