A248751 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

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.

3

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

(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).

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

LINKS

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

Index entries for algebraic numbers, degree 4

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(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((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 *)

PROG

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

CROSSREFS

Cf. A248750, A248752, A102426, A001622.

Sequence in context: A021658 A270859 A248749 * A021193 A010483 A388180

Adjacent sequences: A248748 A248749 A248750 * A248752 A248753 A248754

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, Oct 13 2014

STATUS

approved