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%I #19 Apr 19 2018 10:00:00
%S 1,6,2,8,1,6,0,1,2,9,7,1,8,9,1,7,2,4,8,8,2,1,5,3,1,2,2,5,4,0,9,9,7,3,
%T 2,3,4,9,9,5,1,5,5,7,5,9,2,6,9,7,7,7,6,5,4,1,6,1,6,2,7,9,7,4,6,8,9,9,
%U 0,7,9,1,1,3,1,5,7,6,7,9,1,6,1,6,7,4,8
%N Decimal expansion of Product_{n>=1} (2n/(2n+1))^((-1)^t(n)), where t(n) = A010060(n) is the Thue-Morse sequence.
%C It is an open problem to decide if this number is algebraic.
%C The sequence of partial products P_k = Product_{n=1..2^k-1} (2n/(2n+1))^((-1)^t(n)) converges rapidly to the limit as k increases; e.g., P_28 is correct to more than 100 decimal digits. - _Jon E. Schoenfield_, Aug 17 2014
%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 207.
%D J. Shallit, Number theory and formal languages, in Emerging applications of number theory (Minneapolis, MN, 1996), 547-570, IMA Vol. Math. Appl., 109, Springer, New York, 1999.
%H Jon E. Schoenfield, <a href="/A086744/b086744.txt">Table of n, a(n) for n = 1..100</a>
%e 1.6281601297189...
%t kmax = 28; digits = 100;
%t p[k_] := p[k] = Product[(2n/(2n+1))^(-1)^ThueMorse[n] // N[#, digits+10]&, {n, 2^(k-1), 2^k-1}];
%t RealDigits[Product[Print["p(", k, ") = ", p[k]]; p[k], {k, 1, kmax}], 10, digits][[1]] (* _Jean-François Alcover_, Apr 18 2018, after _Jon E. Schoenfield_'s comments *)
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_, Sep 12 2003
%E More terms from _Vaclav Kotesovec_, Jul 28 2013
%E More terms from _Jon E. Schoenfield_, Aug 17 2014
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