%I #19 Dec 21 2020 07:30:48
%S 5,6,8,6,9,8,9,4,6,2,6,5,4,2,8,5,0,5,9,5,4,9,7,6,7,3,7,0,7,4,4,4,4,6,
%T 5,4,2,9,0,8,5,2,4,5,1,3,8,9,3,5,9,0,2,9,3,1,9,3,4,4,0,4,6,0,1,8,3,5,
%U 3,5,6,3,2,3,0,9,1,2,6,4,0,9,6,1,4,6,4,4,1,1,7,3,0,6,1,4,8,6,0,4,8,0,2,7,2,6,9,4,1,8
%N Decimal expansion of Product_{k>=1} (1 + 1/(-2)^k).
%F Equals Product_{k>=1} 1/(1 + 1/2^(2*k-1)).
%F Equals exp(Sum_{k>=1} A000593(k)/(k*(-2)^k)).
%F From _Peter Bala_, Dec 15 2020: (Start)
%F Constant C = (2/3) - (1/3)*Sum_{n >= 0} (-1)^n * 2^(n^2)/( Product_{k = 1..n+1} 4^k - 1 ).
%F C = Sum_{n >= 0} 1/( Product_{k = 1..n} (-2)^k - 1 ) = 1 - 1/3 - 1/9 + 1/81 + 1/1215 - - + + ... = Sum_{n >= 0} 1/A216206(n).
%F C = 1 + Sum_{n >= 0} (-1/2)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
%F 3*C = 2 - Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
%F 9*C = 5 - Sum_{n >= 0} (-1/8)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
%F 81*C = 46 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
%F 1215*C = 691 + Sum_{n >= 0} (-1/32)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
%F The sequence [1, 2, 5, 46, 691, ...] is the sequence of numerators of the partial sums of the series Sum_{n >= 0} 1/A216206(n). (End)
%e (1 - 1/2) * (1 + 1/2^2) * (1 - 1/2^3) * (1 + 1/2^4) * (1 - 1/2^5) * ... = 0.568698946265428505954976737...
%t RealDigits[QPochhammer[-1, -1/2]/2, 10, 110] [[1]]
%t N[3/QPochhammer[-2, 1/4], 120] (* _Vaclav Kotesovec_, Apr 28 2020 *)
%o (PARI) prodinf(k=1, 1 + 1/(-2)^k) \\ _Michel Marcus_, Apr 28 2020
%Y Cf. A000593, A027637, A048651, A062510, A065446, A079555, A081845, A085011, A100221, A132020, A330862, A216206.
%K nonn,cons
%O 0,1
%A _Ilya Gutkovskiy_, Apr 28 2020