A248751 Decimal expansion of limit of the real part of f(1-i,n)/f(1-i,n+1), where f(x,n) is the n-th Fibonacci polynomial.

5, 2, 9, 0, 8, 5, 5, 1, 3, 6, 3, 5, 7, 4, 6, 1, 2, 5, 1, 6, 0, 9, 9, 0, 5, 2, 3, 7, 9, 0, 2, 2, 5, 2, 1, 0, 6, 1, 9, 3, 6, 5, 0, 4, 9, 8, 3, 8, 9, 0, 9, 7, 4, 3, 1, 4, 0, 7, 7, 1, 1, 7, 6, 3, 2, 0, 2, 3, 9, 8, 1, 1, 5, 7, 9, 1, 8, 9, 4, 6, 2, 7, 7, 1, 1, 4

Offset: 0

Author

Clark Kimberling, Oct 13 2014

Keywords

• nonn
• easy
• cons

Comments

The analogous limit of f(1,n)/f(1,n+1) is the golden ratio (A001622).

Differs from A248749 only in the first digit. - R. J. Mathar, Oct 23 2014

Examples

			limit = 0.52908551363574612516099052379022521061936504...
Let q(x,n) = f(x,n)/f(x,n+1) and c = 1-i.
n   f(x,n)            Re(q(c,n))   Im(q(c,n))
1   1                 1/2          1/2
2   x                 3/5          1/5
3   1 + x^2           1/2          1/4
4   2*x + x^3         8/15         4/15
5   1 + 3*x^2 + x^4   69/130       33/130
Re(q(1-i,11)) = 5021/9490 = 0.5290832...
Im(q(1-i,11)) = 4879/18980 = 0.257060...
		

Links

• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A248750, A248752, A102426, A001622.

Programs

Maple

evalf((sqrt(2+sqrt(5))-1)/2, 120); # Vaclav Kotesovec, Oct 19 2014

Mathematica

z = 300; t = Table[Fibonacci[n, x]/Fibonacci[n + 1, x], {n, 1, z}];
u = t /. x -> 1 - I;
d1 = N[Re[u][[z]], 130]
d2 = N[Im[u][[z]], 130]
r1 = RealDigits[d1]  (* A248751 *)
r2 = RealDigits[d2]  (* A248752 *)

PARI

polrootsreal(4*x^4+8*x^3+2*x^2-2*x-1)[2] \\ Charles R Greathouse IV, Nov 26 2024

Formula

Equals (sqrt(2+sqrt(5))-1)/2. - Vaclav Kotesovec, Oct 19 2014