A374332 a(n) is the numerator of x(n) = (2*x(n-1) + 1/n) mod 1, with x(0) = 0.

0, 0, 1, 1, 11, 1, 7, 64, 289, 1007, 44, 338, 163, 3505, 8297, 44488, 27221, 823117, 993287, 20403983, 26327699, 27713369, 27650353, 315868349, 2488325579, 6016553239, 1399433807, 3562923992, 9142117861, 275160597119, 268889538733, 3968532770473, 114095155444597

Offset: 0

Author

Paolo Xausa, Jul 06 2024

Keywords

• nonn
• frac

Comments

A constant alpha, defined as alpha = Sum_{n >= 1} p(n)/(q(n)*b^n), is b-normal if and only if the associated sequence, defined by x(0) = 0 and x(n) = (b*x(n-1) + p(n)/q(n)) mod 1, is equidistributed in the unit interval.

The present sequence gives the numerators of the associated sequence for alpha = log(2) (where b = 2). See Bailey and Borwein (2005), p. 505 (first example of Theorem 3).

Denominators are given by A374333.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..2000
• David H. Bailey and Jonathan M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the American Mathematical Society, May 2005, Vol. 52, No. 5, pp. 502-514.
• David H. Bailey and Richard E. Crandall, On the Random Character of Fundamental Constant Expansions, Experimental Mathematics, Vol. 10 (2001), Issue 2, pp. 175-190 (preprint draft).
• David H. Bailey and Richard E. Crandall, Random Generators and Normal Numbers, Experimental Mathematics, Vol. 11 (2002), Issue 4, pp. 527-546 (preprint draft).

Crossrefs

Cf. A002162, A374333 (denominators), A374334, A374336.

Programs

Mathematica

Block[{n = 0}, Numerator[NestList[Mod[2*# + 1/++n, 1] &, 0, 50]]]

PARI

x(n) = if (n==0, 0, 2*x(n-1) + 1/n);
a(n) = numerator(frac(x(n))); \\ Michel Marcus, Jul 13 2024

Python

from fractions import Fraction
from itertools import count, islice
def A374332_gen(): # generator of terms
    a = Fraction(0,1)
    for n in count(1):
        yield a.numerator
        a = (2*a+Fraction(1,n)) % 1
A374332_list = list(islice(A374332_gen(),20)) # Chai Wah Wu, Jul 13 2024