%I #16 Dec 26 2021 19:21:17

%S 7,7,3,5,1,6,2,9,0,9,0,8,4,4,5,3,0,4,0,7,3,3,0,2,5,8,5,7,0,7,4,0,1,2,

%T 0,0,3,5,6,7,4,4,4,7,6,2,3,5,0,2,0,7,6,1,2,7,0,2,8,6,4,2,8,7,5,8,7,4,

%U 0,1,5,8,1,7,9,8,7,9,0,1,0,0,5,5,6,8,7

%N Decimal expansion of exp(gamma)/sqrt(2)*Product_{n>=1} ((2n+1)/(2n))^((-1)^t(n)), a probabilistic counting constant, where gamma is Euler's constant and t(n) = A010060(n) the Thue-Morse sequence.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.8 Prouhet-Thue-Morse constant, p. 437.

%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 phi 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Flajolet%E2%80%93Martin_algorithm">Flajolet-Martin algorithm</a>

%F exp(gamma)/(sqrt(2)*A086744).

%e 0.7735162909084453040733025857074...

%t digits = 80; t[n_] := Mod[DigitCount[n, 2, 1], 2]; p[k_] := p[k] = Product[(2*n/(2*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[E^EulerGamma / (Sqrt[2] * Times @@ pp), 10, digits] // First

%Y Cf. A010060, A086744.

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Jun 24 2014

%E A few more digits from _Jon E. Schoenfield_, Oct 13 2014