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%I #12 Dec 26 2021 19:24:11
%S 8,7,1,1,5,7,0,4,6,4,1,4,8,9,3,7,4,1,6,1,7,8,5,7,6,5,6,4,5,9,1,9,1,6,
%T 0,6,2,6,0,3,9,2,3,2,6,3,9,7,5,2,4,1,8,9,1,2,9,0,2,2,7,1,3,8,0,0,9,3,
%U 1,8,2,4,6,6,1,2,4,7,5,4,1,7,0,8,7,8,3
%N Decimal expansion of product_{n>=1} (2n/(2n+1))^((-1)^t(n-1)), a probabilistic counting constant, where t(n) = A010060(n) is the Thue-Morse sequence.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.8 Prouhet-Thue-Morse constant, p. 438.
%H J.-P. Allouche and Jeffrey Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps">The Ubiquitous Prouhet-Thue-Morse Sequence</a>, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16. See the constant Q on page 6.
%H Philippe Flajolet and G. Nigel Martin, <a href="http://www.mathcs.emory.edu/~cheung/papers/StreamDB/Probab/1985-Flajolet-Probabilistic-counting.pdf">Probabilistic counting algorithms for data base applications</a>, Journal of Computer and System Sciences. Vol. 31, No. 2, October 1985, p. 193.
%e 0.871157046414893741617857656459191606260392326397524189129...
%t digits = 60; t[n_] := Mod[DigitCount[n, 2, 1], 2]; Clear[p]; p[1] = 5/6; p[k_] := p[k] = Product[(2*n/(2*n+1))^(-1)^t[n-1], {n, 2^(k-1)+1, 2^k}] // N[#, digits + 40]&; pp = Table[Print["k = ", k]; p[k], {k, 1, 23}]; RealDigits[Times @@ pp, 10, digits] // First
%Y Cf. A010060, A086744, A244256, A248342, A248581.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Oct 09 2014
%E Error beginning at the 15th digit detected by _Jon E. Schoenfield_ and corrected by _Jean-François Alcover_, Oct 22 2014
%E More terms from _Jon E. Schoenfield_, Oct 22 2014