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%I #18 Oct 18 2014 13:33:04
%S 8,1,1,6,8,6,9,2,1,5,4,4,7,7,9,3,1,0,0,8,8,7,6,9,4,5,4,7,7,6,9,1,4,4,
%T 4,0,8,1,1,9,3,4,9,3,5,0,0,9,9,8,5,6,5,4,3,1,2,9,8,3,0,3,7,4,3,7,0,3,
%U 1,6,2,2,9,4,3,9,6,1,1,9,2,1,9,4,3,8,9
%N Decimal expansion of product_{n>=1} (n/(n+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 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 0.811686921544779310088769454776914440811934935...
%t digits = 60; t[n_] := Mod[DigitCount[n, 2, 1], 2]; Clear[p]; p[1] = 3/4; p[k_] := p[k] = Product[(n/(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.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Oct 09 2014
%E Error beginning at the 14th digit detected by _Jon E. Schoenfield_ and corrected by _Jean-François Alcover_, Oct 17 2014
%E A few more digits from _Jon E. Schoenfield_, Oct 17 2014