A229350 Decimal expansion of continued fraction [x(1),x(2),x(3),...], where x(n) = F(n+1)/F(n), where F = A000045 (Fibonacci numbers).

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

Offset

1,2

Comments

Suppose that x(n) is a sequence of positive real numbers with divergent sum. By the Seidel Convergence Theorem, the continued fraction [x(1),x(2),x(3),...] converges.

The fact that F(n+1)/F(n) is the n-th convergent of a continued fraction (specifically, of the golden ratio) exemplifies a certain function f of a positive real variable x: let p(i)/q(i), for i >=0, be the convergents to x; then f(x) is the number [p(0)/q(0), p(1)/q(1), p(2)/q(2), ... ]. For x = golden ratio, f(x) = 1.398598..., f(f(x)) = 1.4903397..., f(f(f(x))) = 1.4995061.... Let L(x) = lim(f(n,x)), where f(0,x) = x, f(1,x) = f(x), and f(n,x) = f(f(n-1,x)). It appears that L(golden ratio) = 3/2.

Links

Table of n, a(n) for n=1..86.

Formula

[1, [1,1], [1,1,1], [1,1,1,1], ... ]. (Here, as in the Name and Example sections, square brackets indicate continued fractions.)

Example

[1, 2/1, 3/2, 5/3, 8/5,...] = [1,2,1,1,27,1,16,670,9,3,2,1,13,1,4,1,1,1...] = 1.3985985...  The first 5 ordinary convergents are 1, 3/2, 4/3, 7/5, 193/138.

Mathematica

z = 500; t = Table[Fibonacci[n + 1]/Fibonacci[n], {n, z}]
r = FromContinuedFraction[t]; c = Convergents[r, z];
Numerator[c]  (* A229348 *)
Denominator[c]  (* A229349 *)
RealDigits[r, 10, 120] (* A229350 *)

Crossrefs

Cf. A229348, A229349, A229351.

Sequence in context: A378021 A199052 A021255 * A070342 A125125 A021719

Adjacent sequences: A229347 A229348 A229349 * A229351 A229352 A229353

Keyword

nonn,cons,easy

Author

Clark Kimberling, Sep 21 2013

Status

approved