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A229352

Denominators of the ordinary convergents of continued fraction [2/1,3/2,4/3,5/4,...].

1, 2, 65, 67, 132, 199, 1923, 9814, 11737, 21551, 54839, 76390, 665959, 1408308, 3482575, 11856033, 133898938, 547451785, 681350723, 1228802508, 1910153231, 3138955739, 8188064709, 19515085157, 125278575651, 144793660808, 414865897267, 559659558075

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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..28.

EXAMPLE

[2/1, 3/2, 4/3, 5/4, ...] = 2.492459746021286... The first 5 ordinary convergents are 2, 5/2, 162/65, 167/67, 329/132.

MATHEMATICA

z = 500; t = Table[(n+1)/n, {n, z}]

r = FromContinuedFraction[t]; c = Convergents[r, z];

Numerator[c] (* A229351 *)

Denominator[c] (* A229352 *)

RealDigits[r, 10, 120] (* A229353 *)

Take[Convergents[FromContinuedFraction[Table[(n+1)/n, {n, 100}]]]//Denominator, 30] (* Harvey P. Dale, Jan 31 2023 *)

CROSSREFS