A248752 Decimal expansion of limit of the imaginary part of f(1-i,n)/f(1-i,n+1), where f(x,n) is the n-th Fibonacci polynomial.

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

Offset

0,1

Comments

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

Links

Table of n, a(n) for n=0..85.

Formula

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

From Wolfdieter Lang, Mar 02 2018: (Start)

Equals (1 - (2 - phi)*sqrt(phi))/2, with phi = A001622.

Equals (1/10)*y*(1 - (1/50)*y^2) with y = A300070. (End)

Example

limit = 0.2570658641216771609085680620527337189037570...
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...

Maple

evalf((1-sqrt(sqrt(5)-2))/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 *)

Crossrefs

Cf. A248750, A248751, A102426, A001622, A300070.

Sequence in context: A231364 A025123 A071791 * A021393 A181583 A010589

Adjacent sequences: A248749 A248750 A248751 * A248753 A248754 A248755

Keyword

nonn,easy,cons,changed

Author

Clark Kimberling, Oct 13 2014

Status

approved