A244331 Number of binary digits in the high-water marks of the terms of the continued fraction of the base-2 Champernowne constant.

0, 1, 3, 9, 23, 53, 115, 241, 495, 1005, 2027, 4073, 8167, 16357, 32739, 65505, 131039, 262109, 524251, 1048537, 2097111, 4194261, 8388563, 16777169

Offset: 1

Author

John K. Sikora, Jun 27 2014

Keywords

• base
• nonn
• more

Comments

Conjecture: partial sums of A296965 (equivalent to observation about A183155 below). - Sean A. Irvine, Jul 16 2022

Links

• John K. Sikora, Table of n, a(n) for n = 1..24
• John K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT]
• John K. Sikora, Number of binary digits of the first 98093504 terms of the continued fraction of the base 2 Champernowne Constant (240 MB zipped)

Crossrefs

Cf. A066717, A066718, A244330.

Programs

Ruby

puts (4..24).collect{|n| 2**n-2*n+1}

Ruby

puts (4..24).collect {|n| (1..n).inject(0) {|sum, m| sum+m*2**(m-1)}-n-2*((1..(n-1)).inject(0) {|sum1, m1| sum1+m1*2**(m1-1)}-(n-1))-3*n+4}

Formula

It appears that for n >= 4, a(n) = 2^n - 2*n + 1 = A183155(n-1).

Also it appears that if we define NCD(N) = (Sum_{m=1..N} m*2^(m-1)) - N, then for n >= 4, a(n) = NCD(n) - 2*NCD(n-1) - 3*n + 4.