A229348 Numerators of the ordinary convergents of continued fraction [x(1),x(2),x(3),...], where x(n) = F(n+1)/F(n), F = A000045 (Fibonacci numbers).

1, 3, 4, 7, 193, 200, 3393, 2273510, 20464983, 63668459, 147801901, 211470360, 2896916581, 3108386941, 15330464345, 18438851286, 33769315631, 52208166917, 347018317133, 399226484050, 4738509641683, 9876245767416, 14614755409099, 39105756585614

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.

Links

Table of n, a(n) for n=1..24.

Example

[1, 2/1, 3/2, 5/3, 8/5,...] = 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. A229349, A229350, A229351.

Sequence in context: A100978 A059077 A042707 * A215758 A011215 A093592

Adjacent sequences: A229345 A229346 A229347 * A229349 A229350 A229351

Keyword

nonn,frac,easy

Author

Clark Kimberling, Sep 21 2013

Status

approved