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%I #20 Oct 14 2014 04:23:47
%S 2,3,0,2,5,6,6,1,3,7,1,6,3,6,3,0,5,0,4,2,9,3,5,8,1,2,5,1,2,4,1,8,1,5,
%T 5,9,3,6,4,0,1,2,3,5,9,6,5,0,5,9,1,1,0,1,1,4,6,1,6,7,0,9,0,4,1,3,9,7,
%U 6,6,3,3,9,9,5,0,2,6,2,4,3,2,9,0,9,9,2
%N Decimal expansion of product_{n>=1} (n/(n+1))^((-1)^t(n)), 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 2.302566137163630504293581251241815593640123596505911...
%t digits = 80; t[n_] := Mod[DigitCount[n, 2, 1], 2]; p[k_] := p[k] = Product[(n/(n+1))^((-1)^t[n]), {n, 2^k, 2^(k+1)-1}] // N[#, digits + 20]&; pp = Table[Print["k = ", k]; p[k], {k, 0, 24}]; RealDigits[ Times @@ pp , 10, digits] // First
%Y Cf. A010060, A086744, A244256.
%K nonn,cons
%O 1,1
%A _Jean-François Alcover_, Oct 08 2014
%E A few more digits from _Jon E. Schoenfield_, Oct 13 2014