A086744 Decimal expansion of Product_{n>=1} (2n/(2n+1))^((-1)^t(n)), where t(n) = A010060(n) is the Thue-Morse sequence.

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, 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, 0, 7, 9, 1, 1, 3, 1, 5, 7, 6, 7, 9, 1, 6, 1, 6, 7, 4, 8

Offset

1,2

Comments

It is an open problem to decide if this number is algebraic.

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

References

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 207.

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.

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..100

Example

1.6281601297189...

Mathematica

kmax = 28; digits = 100;
p[k_] := p[k] = Product[(2n/(2n+1))^(-1)^ThueMorse[n] // N[#, digits+10]&, {n, 2^(k-1), 2^k-1}];
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 *)

Crossrefs

Sequence in context: A368669 A270138 A177889 * A242301 A256129 A019692

Adjacent sequences: A086741 A086742 A086743 * A086745 A086746 A086747

Keyword

nonn,cons

Author

N. J. A. Sloane, Sep 12 2003

Extensions

More terms from Vaclav Kotesovec, Jul 28 2013

More terms from Jon E. Schoenfield, Aug 17 2014

Status

approved