TOPICS
Search

Catalan's Constant Continued Fraction


Catalan's constant continued fraction binary plot

The simple continued fraction representations for Catalan's constant K is [0, 1, 10, 1, 8, 1, 88, 4, 1, 1, ...] (OEIS A014538). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.

Record computations are summarized below.

termsdateby
970278792Jul. 20, 2013E. Weisstein
4851389025Aug. 7, 2013E. Weisstein
CatalansConstantContinuedFractionFirstOccurrences

The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 1, 13, 14, 7, 45, 36, 10, 4, 21, 2, ... (OEIS A196461; illustrated above). The smallest number not occurring in the first 4851389025 terms of the continued fraction are 31516, 31591, 32600, 32806, 33410, ... (E. Weisstein, Aug. 8, 2013).

The cumulative largest terms in the continued fraction are 0, 1, 10, 88, 322, 330, 1102, 6328, ... (OEIS A099789), which occur at positions 0, 1, 2, 6, 105, 284, 747, 984, 2230, 5377, ... (OEIS A099790).

CatalanKhinchinLevy

Let the continued fraction of K be denoted [a_0;a_1,a_2,...] and let the denominators of the convergents be denoted q_1, q_2, ..., q_n. Then plots above show successive values of a_1^(1/1), (a_1a_2)^(1/2), (a_1a_2...a_n)^(1/n), which appear to converge to Khinchin's constant (left figure) and q_n^(1/n), which appear to converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.

The Engel expansion of K is given by 2, 2, 2, 4, 4, 5, 5, 12, 13, 41, 110, ... (OEIS A054543).


See also

Catalan's Constant, Catalan's Constant Digits

Explore with Wolfram|Alpha

References

Sloane, N. J. A. Sequences A014538, A054543, A099789, A099790, and A196461 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Catalan's Constant Continued Fraction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CatalansConstantContinuedFraction.html

Subject classifications