A374336 a(n) is the numerator of x(n) = (2*x(n-1) + c(n)) mod 1, where c(n) = 1/n if n is a power of 3 and 0 otherwise, with x(0) = 0.

0, 0, 0, 1, 2, 1, 2, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 13, 26, 25, 23, 19, 11, 22, 17, 7, 14, 1, 2, 4, 8, 16, 5, 10, 20, 13, 26, 25, 23, 19, 11, 22, 17, 7, 14, 1, 2, 4, 8, 16, 5, 10, 20, 13, 26, 25, 23, 19, 11, 22, 17, 7, 14, 1, 2, 4

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 (where b = 2) for alpha_0 = Sum_{n >= 1} 1/((3^n)*2^(3^n)) = A192014. See Bailey and Borwein (2005), pp. 505-506 (third example of Theorem 3). They show that alpha_0, as well as any constant defined as Sum_{n >= 1} 1/((3^n)*2^(3^n+r_n)) (where r_n is the n-th binary digit of the real number r in the [0,1) interval), is 2-normal and transcendental.

Bailey and Borwein also note that terms follow a pattern of triply repeating segments, each of length 2*3^m and containing all integers relative prime to and less than 3^(m+1).

Denominators are given by A365458.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• 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.

Crossrefs

Cf. A192014, A374332, A374334, A365458 (denominators).

Programs

Mathematica

Block[{n = 0}, Numerator[NestList[Mod[2*# + If[IntegerQ[Log[3, ++n]], 1/n, 0], 1] &, 0, 100]]]