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 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
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.
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.
EXAMPLE
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.
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
KEYWORD
cofr,nonn
AUTHOR
Hans Havermann, Jan 03 2004
STATUS
approved