A229348 - OEIS

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

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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.