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A248582 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. 0

%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

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Last modified March 28 14:13 EDT 2024. Contains 371254 sequences. (Running on oeis4.)