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

0,1

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

Links

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

Formula

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

Example

limit = 0.52908551363574612516099052379022521061936504...
Let q(x,n) = f(x,n)/f(x,n+1) and c = 1-i.
n   f(n,x)                 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    2x + x^3              8/15             4/15
5    1 + 3 x^2 + x^4       69/130           33/130
Re(q(11,1+i) = 5021/9490 = 0.5290832...
Im(q(11,1+i) = 4879/18980 = 0.275060...

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 *)

Crossrefs

Cf. A248750, A248752, A102426, A001622.

Sequence in context: A021658 A270859 A248749 * A021193 A010483 A003671

Adjacent sequences: A248748 A248749 A248750 * A248752 A248753 A248754

Keyword

nonn,easy,cons

Author

Clark Kimberling, Oct 13 2014

Status

approved