A143534 Number of decimal digits in the high-water marks of the terms of the continued fraction of the (base-10) Champernowne constant.

0, 1, 1, 6, 166, 2504, 33102, 411100, 4911098, 57111096, 651111094, 7311111092

Offset: 1

Author

Eric W. Weisstein, Aug 22 2008

Keywords

• nonn
• more
• base

Links

• J. K. Sikora, On the High Water Mark Convergents of Champernowne's Constant in Base Ten, arXiv:1210.1263 [math.NT]
• Eric Weisstein's World of Mathematics, Champernowne Constant Continued Fraction

Crossrefs

Cf. A038705 (position of the incrementally largest term in continued fraction for Champernowne constant).

Cf. A143533 (another version of A038705).

Programs

Ruby

puts (4..13).collect {|n| (1..(n-3)).inject(0) {|sum, m| sum+9*m*10**(m-1)}+3-n-2*((1..(n-4)).inject(0) {|sum1, m1| sum1+9*m1*10**(m1-1)}+3-(n-1))-3*(n-2)+4} # John K. Sikora, Aug 25 2012

Formula

It appears that: For N>=3, define NCD(N)=3-N+(sum{m=1..(N-3), m>0} 9*m*10^(m-1)); then for n>=4, a(n) = NCD(n) - 2*NCD(n-1) - 3*(n-2) + 4. - John K. Sikora, Aug 25 2012

Extensions

a(11) = 651111094 (from Mark Sofroniou), Eric W. Weisstein, Sep 04 2008

a(12) = 7311111092 from Eric W. Weisstein, Jun 29 2013