Champernowne Constant Continued Fraction

The Champernowne constant continued fraction is the regular continued fraction of the Champernowne constant. Its first few terms are [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 45754...10987, 6, 1, 1, 21, ...] (OEIS A030167), and the numbers of decimal digits in these terms, assigning the initial zero a digit count of 0, are 0, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 166, 1, ... (OEIS A143532). E. W. Weisstein computed 82,328 terms of the continued fraction on Jun. 30, 2013 using the Wolfram Language.

In the continued fraction , count the initial 0 as position 1, so has position . First occurrences of the terms 1, 2, 3, ... are at positions 4, 28, 13, 9, 93, 20, 31, 2, 3, 339, 71, 126, 107, ... (OEIS A038706). In the reported 82,328-term computation, 188 was the smallest positive integer not encountered, so any first occurrence has position greater than 82,328.

The continued fraction contains sporadic very large terms, making the continued fraction difficult to calculate. However, the sizes of the continued fraction high-water marks display apparent patterns (Sikora 2012). Large terms greater than occur at positions 5, 19, 41, 102, 163, 247, 358, 460, ... and have 6, 166, 2504, 140, 33102, 109, 2468, 136, ... decimal digits, respectively.

The high-water marks in the continued fraction occur at coefficient indices , 1, 2, 4, 18, 40, 162, 526, 1708, 4838, 13522, 34062, ... (OEIS A143533; Sikora 2012), which have 0, 1, 1, 6, 166, 2504, 33102, 411100, 4911098, 57111096, 651111094, 7311111092, ... (OEIS A143534) decimal digits, respectively, using the same initial-zero convention. Sikora (2012, pp. 5 and 8) predicted the twelfth digit count; OEIS A143534 subsequently attributed that value to E. W. Weisstein (Jun. 29, 2013). Sikora (2012, p. 8) conjectured that the number of decimal digits in the th high-water mark for is given by

(1)

where, for an integer ,

			(2)
			(3)

in agreement with the listed values through .