A248749 Decimal expansion of limit of the real part of f(1+i,n), where f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1).

1, 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

Offset: 1

Author

Clark Kimberling, Oct 13 2014

Keywords

• nonn
• easy
• cons

Comments

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.

Examples

			limit = 1.52908551363574612516099052379022521061936504983890974314077117...
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...
		

Crossrefs

Cf. A248750, A046854.

Programs

Maple

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

Mathematica

$RecursionLimit = Infinity; $MaxExtraPrecision = Infinity;
f[x_, n_] := x + 1/f[x, n - 1]; f[x_, 1] = 1; t = Table[Factor[f[x, n]], {n, 1, 12}]; u = t /. x -> I + 1; {Re[u], Im[u]}
{N[Re[u], 12], N[Im[u], 12]}
t = Table[Factor[f[x, n]], {n, 1, 300}]; u = t /. x -> I + 1;
r1 = N[Re[u][[300]], 130]
r2 = N[Im[u][[300]], 130]
d1 = RealDigits[r1]  (* A248749 *)
d2 = RealDigits[r2]  (* A248750 *)

Formula

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