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
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