A244333 - OEIS

%I #22 Aug 07 2014 03:25:24

%S 0,1,1,4,5,33,139,515,1809,6181,20759,68871,226333,738089,2391459,

%T 7705867,24711977,78918957,251105839

%N Number of ternary digits in the high-water marks of the terms of the continued fraction of the base 3 Champernowne constant (A077772).

%H John K. Sikora, <a href="/A244333/b244333.txt">Table of n, a(n) for n = 1..19</a>

%H J. 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 J. K. Sikora, <a href="https://drive.google.com/file/d/0B2oQudZQvHfSOXF6WVkzWVVueWs/edit?usp=sharing">Number of ternary digits of the first 2982556 terms of the continued fraction of the base 3 Champernowne Constant (7 MB zipped)</a>

%F It appears that: Define NCD(N)=3-N+(sum{m=1..(N-3)} 2*m*3^(m-1)); then for n>=5, a(n) = NCD(n)-2*NCD(n-1)-3*(n-2)+4.

%o (Ruby) puts (5..19).collect {|n| (1..(n-3)).inject(0) {|sum, m| sum+2*m*3**(m-1)}+3-n-2*((1..(n-4)).inject(0) {|sum1, m1| sum1+2*m1*3**(m1-1)}+3-(n-1))-3*(n-2)+4}

%Y Cf. A077772 (Continued fraction expansion of the ternary Champernowne constant.)

%Y Cf. A244757 (Incrementally largest terms in the continued fraction for the base 3 Champernowne constant.)

%Y Cf. A244332 (Position of the incrementally largest term in the continued fraction for the base 3 Champernowne constant.)

%K base,nonn,more

%O 1,4

%A _John K. Sikora_, Jun 27 2014