A248749 -id:A248749 - cp's OEIS frontend

A248750 Decimal expansion of limit of the imaginary part of f(1+i,n), where f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1).

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

Offset: 0

Clark Kimberling, Oct 13 2014

See A046854 for a triangle of coefficients of the numerators and denominators of f(x,n). Note that the limit of f(1,n) is the golden ratio.

			0.742934135878322839091431937947266281096242992001186505475869206219...
n   f(x,n)                                 Re(f(1+i,n))  Im(f(1+i,n))
0   1                                      1             0
1   1 + x                                  2             1
2   (1 + x + x^2)/(1 + x)                  7/5           4/5
3   (1 + 2*x + x^2 + x^3)/(1 + x + x^2)    20/13         9/13
Re(f(1+i,10)) = 815/533 = 1.529080...
Im(f(1+i,10)) = 396/533 = 0.742964...

Index entries for algebraic numbers, degree 4.

Cf. A248749, A046854, A248751, A248752, A001622. A300072.

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

$RecursionLimit = Infinity; $MaxExtraPrecision = Infinity;
z = 20; (* For more accuracy, increase z *)
f[x_, n_] := x + 1/f[x, n - 1];
f[x_, 1] = 1; t = Table[Factor[f[x, n]], {n, 1, z}];
u = t /. x -> I + 1; t = Table[Factor[f[x, n]], {n, 1, z}]; u = t /. x -> I + 1;
r1 = N[Re[u][[z]], 130]
r2 = N[Im[u][[z]], 130]
d1 = RealDigits[r1]  (*A248749*)
d2 = RealDigits[r2]  (*A248750*)

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 = -A300072. (End)

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.

Original entry on oeis.org

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

Clark Kimberling, Oct 13 2014

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

			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...

Index entries for algebraic numbers, degree 4

Cf. A248750, A248752, A102426, A001622.

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

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 *)

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

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

cp's OEIS Frontend

A248750 Decimal expansion of limit of the imaginary part of f(1+i,n), where f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1).

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