A374334 a(n) is the numerator of x(n) = (16*x(n-1) + (120*n^2 - 89*n + 16)/(512*n^4 - 1024*n^3 + 712*n^2 - 206*n + 21)) mod 1, with x(0) = 0.

0, 2, 1076, 188663, 106894973, 32442016954, 16143697977964, 43667396600461261, 82482175187690988496, 80845733759021750791, 209616749220518838502, 48891577015658186678698, 60882892596227901210360094, 108196850082040258114673507582, 189145139720511629801253759599798

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 (with b = 2) for alpha = Pi. See Bailey and Borwein (2005), p. 505 (second example of Theorem 3). In the same paper, on p. 513, they conjecture that, for n >= 1, y(n) = floor(16*x(n)) = A062964(n+1). See also Bailey and Crandall (2001), p. 176.

Denominators are given by A374335.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..375
• 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).

Crossrefs

Cf. A000796, A062964, A374332, A374335 (denominators), A374336, A374580, A374607.

Programs

Mathematica

Block[{n = 0}, Numerator[NestList[Mod[16*# + (120*(++n)^2 - 89*n + 16)/(512*n^4 - 1024*n^3 + 712*n^2 - 206*n + 21), 1] &, 0, 20]]]