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%I #22 Jul 14 2024 17:59:06
%S 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,
%T 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,
%U 8,16,5,10,20,13,26,25,23,19,11,22,17,7,14,1,2,4
%N 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.
%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 (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.
%C 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).
%C Denominators are given by A365458.
%H Paolo Xausa, <a href="/A374336/b374336.txt">Table of n, a(n) for n = 0..10000</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.
%t Block[{n = 0}, Numerator[NestList[Mod[2*# + If[IntegerQ[Log[3, ++n]], 1/n, 0], 1] &, 0, 100]]]
%Y Cf. A192014, A374332, A374334, A365458 (denominators).
%K nonn,frac
%O 0,5
%A _Paolo Xausa_, Jul 06 2024
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