A171681 a(n) = F(2n+1)^3 - F(3n)^2 - F(6n-2), where the F(i) are Fibonacci numbers.

1, 6, 54, 857, 15058, 269394, 4831929, 86699846, 1555750918, 27916779057, 500946173586, 8989114087586, 161303106727729, 2894466805243782, 51939099383032278, 932009322077220809, 16724228697975221074

Offset

1,2

Comments

The ratio of two consecutive terms of this sequence, as n goes to infinity, is phi^6 = 8*phi+5 = 9+4*sqrt(5) where phi is the golden ratio=1.618...

Links

G. C. Greubel, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (20,-35,-35,20,-1).

Formula

a(n) = 20*a(n-1) - 35*a(n-2) - 35*a(n-3) + 20*a(n-4) - a(n-5). - R. J. Mathar, Nov 23 2010

G.f.: x*(1-14*x-31*x^2+22*x^3-2*x^4) / ((1+x)*(x^2-3*x+1)*(x^2-18*x+1)).

a(n+1) = (-2*(-1)^n + A134493(n+1) + 3*A001519(n+2))/5. - R. J. Mathar, Nov 23 2010

Example

d(3) = 54 since F(7)^3 = F(9)^2 + F(16) + 54.

Mathematica

Table[(1/5)*(3*Fibonacci[2*n + 1] + Fibonacci[6*n - 5] + 2*(-1)^n), {n, 1, 10}] (* G. C. Greubel, Apr 18 2016 *)
LinearRecurrence[{20, -35, -35, 20, -1}, {1, 6, 54, 857, 15058}, 20] (* Harvey P. Dale, Dec 15 2017 *)

Crossrefs

Sequence in context: A138434 A360545 A217238 * A267837 A049037 A047681

Adjacent sequences: A171678 A171679 A171680 * A171682 A171683 A171684

Keyword

nonn,easy

Author

Carmine Suriano, Dec 15 2009

Extensions

Simplified the definition. - N. J. A. Sloane, Nov 24 2010

Status

approved