Champernowne Constant Continued Fraction

The first few terms in the continued fraction of the Champernowne constant 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 number of decimal digits in these terms 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  terms of the continued fraction on Jun. 30, 2013 using the Wolfram Language.

First occurrences of the terms 1, 2, 3, ... in the continued fraction  occur at , 28, 13, 9, 93, 20, 31, 2, 3, 339, 71, 126, 107, ... (OEIS A038706). The smallest unknown value is 188, which has .

The continued fraction contains sporadic very large terms, making the continued fraction difficult to calculate. However, the size 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, ... digits, respectively.

The high-water marks in terms of the continued fraction occur for terms 0, 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; Sikora 2012) decimal digits, respectively. Sikora (2012) conjectured that the number of decimal digits in the th high-water mark for  are given by

(1)

where

			(2)
			(3)

in agreement with known calculated values up to .

REFERENCES:

Havermann, H. "Numbers of Digits in Some Champernowne-Continued-Fraction Terms." https://odo.ca/~haha/cfcd.html.

 Rytin, M. "Champernowne Constant and Its Continued Fraction Expansion." https://library.wolfram.com/infocenter/MathSource/2876/.

Sikora, J. K. "On the High Water Mark Convergents of Champernowne's Constant in Base Ten." 3 Oct 2012. https://arxiv.org/abs/1210.1263.

Sloane, N. J. A. Sequences A030167, A030190, A033307, A038706, A054635, A058935, A066716, A066717, A077771, A077772, A143532, A143533, and A143534 in "The On-Line Encyclopedia of Integer Sequences."