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

Author

Carmine Suriano, Dec 15 2009

Keywords

• nonn
• easy

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...

Examples

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

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).

Programs

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 *)

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

Extensions

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