Trott Constants

A Trott constant is a real number whose decimal expansion matches the coefficients of a specified continued fraction. In the original examples, the initial integer part 0 is omitted from the digit stream. The digits are read either from the partial denominators of a simple continued fraction allowing zeros or alternately from the partial numerators and partial denominators of a generalized continued fraction.

The first numerical candidate has decimal prefix (OEIS A039662) and proposed simple continued fraction representation , allowing zero partial denominators (Finch 2003, p. 443). Trott posed the matching problem in 1999 (Allaart et al. 2023). Applying the extension method of Schoenfield (2010), Schoenfield (2015) extended the initial 52 digits to 639 terms. The finite continued fraction and the corresponding decimal concatenation agree through 469 digits. The next extension step would require more than consecutive term pairs (9, 0) (Schoenfield 2015).

The second numerical candidate is (OEIS A091694; Trott 2004, p. 70), with proposed matching non-simple generalized continued fraction

The third numerical candidate is (OEIS A113307; M. Trott, pers. comm., Oct. 24, 2005), with proposed matching non-simple generalized continued fraction

Trott (2006) described these searches as experimental. Finite agreement does not establish the existence or uniqueness of an infinite matching expansion in these original conventions.

A later convention concatenates the full decimal digit strings of the positive integer partial denominators of a regular continued fraction, allowing an individual partial denominator to have several digits. Under this convention there are uncountably many Trott numbers in base 10 (Allaart et al. 2023).