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Catalan's constant approximations

Approximations to Catalan's constant include(1)(2)(3)(4)(5)(6)(M. Hudson, pers. comm., Nov. 19, 2004), where is the golden ratio, which are good to 4, 5, 6, 6, 7, 7, and 9 digits, respectively.Other approximations include(7)(8)(K. Hammond, pers. comm., Dec. 31, 2005), where is the golden ratio, which are good to 5 and 9 digits, respectively.

Catalan's constant digits

Based on methods developer in collaboration with M. Leclert, Catalan (1865) computed the constant(OEIS A006752) now known as Catalans' constant to 9 decimals. In 1867, M. Bresse subsequently computed to 24 decimal places using a technique from Kummer. Glaisher evaluated to 20 (Glaisher 1877) and subsequently 32 decimal digits (Glaisher 1913). Catalan's constant was computed to decimal digits by A. Roberts on Dec. 13, 2010 (Yee).The Earls sequence (starting position of copies of the digit ) for Catalan's constant is given for , 2, ... by 2, 107, 1225, 596, 32187, 185043, 20444527, 92589355, 3487283621, ... (OEIS A224819).-constant primes occur for 52, 276, 25477, ... (OEIS A118328) digits.It is not known if is normal, but the following table giving the counts of digits in the first terms shows that the decimal digits are very uniformly distributed up to at least .OEIS101000A22461506989769828996209997849998686999960671A2246162189410399832996971000293100038131000063052A22469601093980100781001681001789100051221000008063A22470607104101498599958099967299956761000014834A22471711110796110051100074100016599953771000018715A2247743108910031006210005399996599993091000007776A22477511278985998610020199871210000674999988167A2248160111241032100281000831000510100038631000005768A2248170310210581019210035299929899974371000008639A224818312111952100841001729998121000004399992436The..

Catalan's constant continued fraction

The simple continued fraction representations for Catalan's constant 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.termsdatebyJul. 20, 2013E. WeissteinAug. 7, 2013E. WeissteinThe 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 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).Let the continued fraction..

Catalan's constant

Catalan's constant is a constant that commonly appears in estimates of combinatorial functions and in certain classes of sums and definite integrals. It is usually denoted (this work), (e.g., Borwein et al. 2004, p. 49), or (Wolfram Language).Catalan's constant may be defined by(1)(Glaisher 1877, who however did not explicitly identify the constant in this paper). It is not known if is irrational.Catalan's constant is implemented in the WolframLanguage as Catalan.The constant is named in honor of E. C. Catalan (1814-1894), who first gave an equivalent series and expressions in terms of integrals. Numerically,(2)(OEIS A006752). can be given analytically by the following expressions(3)(4)(5)where is the Dirichlet beta function, is Legendre's chi-function, is the Glaisher-Kinkelin constant, and is the partial derivative of the Hurwitz zeta function with respect to the first argument.Glaisher (1913) gave(6)(Vardi..

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