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Number of binary digits in the high-water marks of the terms of the continued fraction of the base-2 Champernowne constant.
3

%I #32 Jul 16 2022 00:54:44

%S 0,1,3,9,23,53,115,241,495,1005,2027,4073,8167,16357,32739,65505,

%T 131039,262109,524251,1048537,2097111,4194261,8388563,16777169

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

%C Conjecture: partial sums of A296965 (equivalent to observation about A183155 below). - _Sean A. Irvine_, Jul 16 2022

%H John K. Sikora, <a href="/A244331/b244331.txt">Table of n, a(n) for n = 1..24</a>

%H John K. Sikora, <a href="http://arxiv.org/abs/1408.0261">Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases</a>, arXiv:1408.0261 [math.NT]

%H John K. Sikora, <a href="https://drive.google.com/file/d/0B2oQudZQvHfSN1hYVjBWdGQ4cVU/edit?usp=sharing">Number of binary digits of the first 98093504 terms of the continued fraction of the base 2 Champernowne Constant (240 MB zipped)</a>

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

%F 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.

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

%o (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}

%Y Cf. A066717, A066718, A244330.

%K base,nonn,more

%O 1,3

%A _John K. Sikora_, Jun 27 2014