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...

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

Offset: 0

Author

Hans Havermann, Jan 03 2004

Keywords

• cofr
• nonn

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 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

Examples

			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.
		

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. 417-431.
• T. Wieting, A Khinchin Sequence, Proc. Amer. Math. Soc., 136 (2008), 815-824.

Crossrefs

Cf. A002210, A087491-A087500, A030101.

Programs

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 *)

Formula

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.

a(n) = floor(1/(2^(A030101(n)/A062383(n))-1)) for n>0. - Andrey Zabolotskiy, Feb 20 2017