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%I #21 Feb 20 2017 11:46:54
%S 0,2,5,1,11,1,3,1,22,2,4,1,7,1,2,1,45,2,4,1,8,1,3,1,14,1,3,1,6,1,2,1,
%T 91,2,4,1,9,1,3,1,17,2,3,1,6,1,2,1,30,2,4,1,7,1,2,1,12,1,3,1,5,1,2,1,
%U 184,2,5,1,10,1,3,1,20,2,4,1,6,1,2,1,36,2,4
%N Continued fraction elements constructed out of a van der Corput discrepancy sequence. Interpreted as such, it is the simple continued fraction of 0.461070495956719519354149869336699687678...
%C The authors of On the Khintchine Constant posit that the geometric mean of the sequence (interpreted as a simple continued fraction expansion) is Khinchin's constant "on the idea that the discrepancy sequence is in a certain sense equidistributed."
%C That conjecture has been proven by Wieting. Moreover, the r-th power mean of the sequence (except a(0)=0, of course) also converges to the corresponding constant K_r for any real r<1. - _Andrey Zabolotskiy_, Feb 20 2017
%H Andrey Zabolotskiy, <a href="/A089618/b089618.txt">Table of n, a(n) for n = 0..10000</a>
%H D. Bailey, J. Borwein, & R. Crandall, <a href="http://crd.lbl.gov/~dhbailey/dhbpapers/khinchine.pdf">On the Khintchine constant</a>, Mathematics of Computation 66:217 (January 1997), pp. 417-431.
%H T. Wieting, <a href="https://doi.org/10.1090/S0002-9939-07-09202-7">A Khinchin Sequence</a>, Proc. Amer. Math. Soc., 136 (2008), 815-824.
%F a(n) = integer part of 1/(2^b(n)-1) where b(n) = digit-reversal of binary of (positive integer) n, preceded by a decimal point and converted (from base 2) to base 10; initial term, a(0), is defined as 0.
%F a(n) = floor(1/(2^(A030101(n)/A062383(n))-1)) for n>0. - _Andrey Zabolotskiy_, Feb 20 2017
%e 40 is 101000 in base 2, so b(40) = 0.078125 (the equivalent of binary 0.000101), 1/(2^0.078125-1) is approximately 17.97 and a(40) is the integer part of this: 17.
%t a[n_] := (m = IntegerDigits[n, 2]; l = Length[m]; s = "2^^."; Do[s = s <> ToString[m[[i]]], {i, l, 1, -1}]; Floor[1/(2^ToExpression[s]-1)]); Prepend[Table[a[i], {i, 1, 120}], 0]
%t a[n_] := If[n==0, 0, Floor[1 / (2^FromDigits[{Reverse[IntegerDigits[n,2]],0},2] - 1)]]; (* _Andrey Zabolotskiy_, Feb 20 2017 *)
%Y Cf. A002210, A087491-A087500, A030101.
%K cofr,nonn
%O 0,2
%A _Hans Havermann_, Jan 03 2004
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