

A089618


Continued fraction elements constructed out of a van der Corput discrepancy sequence. Interpreted as such, it is the simple continued fraction of 0.461070495956719519354149869336699687678...


1



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, 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, 184, 2, 5, 1, 10, 1, 3, 1, 20, 2, 4, 1, 6, 1, 2, 1, 36, 2, 4
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OFFSET

0,2


COMMENTS

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."
That conjecture has been proven by Wieting. Moreover, the rth 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


LINKS

Andrey Zabolotskiy, Table of n, a(n) for n = 0..10000
D. Bailey, J. Borwein, & R. Crandall, On the Khintchine constant, Mathematics of Computation 66:217 (January 1997), pp. 417431.
T. Wieting, A Khinchin Sequence, Proc. Amer. Math. Soc., 136 (2008), 815824.


FORMULA

a(n) = integer part of 1/(2^b(n)1) where b(n) = digitreversal 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.
a(n) = floor(1/(2^(A030101(n)/A062383(n))1)) for n>0.  Andrey Zabolotskiy, Feb 20 2017


EXAMPLE

40 is 101000 in base 2, so b(40) = 0.078125 (the equivalent of binary 0.000101), 1/(2^0.0781251) is approximately 17.97 and a(40) is the integer part of this: 17.


MATHEMATICA

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]
a[n_] := If[n==0, 0, Floor[1 / (2^FromDigits[{Reverse[IntegerDigits[n, 2]], 0}, 2]  1)]]; (* Andrey Zabolotskiy, Feb 20 2017 *)


CROSSREFS

Cf. A002210, A087491A087500, A030101.
Sequence in context: A085358 A146104 A120235 * A207629 A207614 A156067
Adjacent sequences: A089615 A089616 A089617 * A089619 A089620 A089621


KEYWORD

cofr,nonn


AUTHOR

Hans Havermann, Jan 03 2004


STATUS

approved



