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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. 7
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 (list; constant; graph; refs; listen; history; text; internal format)
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(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((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

AUTHOR

Clark Kimberling, Oct 13 2014

STATUS

approved

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Last modified November 28 12:22 EST 2021. Contains 349401 sequences. (Running on oeis4.)