A171680 a(n) = F(2*n)^3 - F(3*n-1)^2 - F(6*n-8).

1, -1, 16, 353, 6535, 117764, 2114521, 37946999, 680940352, 12219002585, 219261167071, 3934482164084, 70601418203761, 1266891046596143, 22733437423387120, 407934982581860369, 7320096249069704311, 131353797500724143204, 2357048258764099246537

Offset

1,3

Comments

Previous name was: If a(n) is a term of this sequence, it represents the remainder of the expression of the cube of a Fibonacci number in terms of a square of a Fibonacci number and another Fibonacci number; if F(n) is the n-th Fibonacci number, then F(2*n)^3 = F(3*n-1)^2 + F(6*n-8) + a(n).

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

Links

Colin Barker, Table of n, a(n) for n = 1..700

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

Formula

From Colin Barker, Mar 13 2016: (Start)

a(n) = fibonacci(2*n)^3-fibonacci(3*n-1)^2-fibonacci(6*n-8).

a(n) = 20*a(n-1)-35*a(n-2)-35*a(n-3)+20*a(n-4)-a(n-5) for n>5.

G.f.: x*(1-21*x+71*x^2+33*x^3-20*x^4) / ((1+x)*(1-18*x+x^2)*(1-3*x+x^2)).

(End)

Example

a(4) = 353 since F(8)^3 = F(11)^2 + F(16) + 353.

Prog

(PARI) vector(26, n, fibonacci(2*n)^3-fibonacci(3*n-1)^2-fibonacci(6*n-8)) \\ Colin Barker, Mar 13 2016
(PARI) Vec(x*(1-21*x+71*x^2+33*x^3-20*x^4)/((1+x)*(1-18*x+x^2)*(1-3*x+x^2)) + O(x^25)) \\ Colin Barker, Mar 13 2016

Crossrefs

Cf. A000045.

Sequence in context: A010368 A053103 A223032 * A208706 A278622 A155122

Adjacent sequences: A171677 A171678 A171679 * A171681 A171682 A171683

Keyword

sign,easy

Author

Carmine Suriano, Dec 15 2009

Extensions

New name base on Colin Barker's formula from Joerg Arndt, Mar 13 2016

Status

approved