%I #30 Jul 14 2024 15:15:40

%S 0,2,1076,188663,106894973,32442016954,16143697977964,

%T 43667396600461261,82482175187690988496,80845733759021750791,

%U 209616749220518838502,48891577015658186678698,60882892596227901210360094,108196850082040258114673507582,189145139720511629801253759599798

%N 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.

%C 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.

%C 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.

%C Denominators are given by A374335.

%H Paolo Xausa, <a href="/A374334/b374334.txt">Table of n, a(n) for n = 0..375</a>

%H David H. Bailey and Jonathan M. Borwein, <a href="https://www.ams.org/notices/200505/fea-borwein.pdf">Experimental Mathematics: Examples, Methods and Implications</a>, Notices of the American Mathematical Society, May 2005, Vol. 52, No. 5, pp. 502-514.

%H David H. Bailey and Richard E. Crandall, <a href="https://doi.org/10.1080/10586458.2001.10504441">On the Random Character of Fundamental Constant Expansions</a>, Experimental Mathematics, Vol. 10 (2001), Issue 2, pp. 175-190 (<a href="https://www.davidhbailey.com/dhbpapers/baicran.pdf">preprint draft</a>).

%t 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]]]

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

%K nonn,frac

%O 0,2

%A _Paolo Xausa_, Jul 06 2024