A083864 Decimal expansion of Product_{k>=0} (1 - 1/(2^k+1)).

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

Offset

0,1

Comments

c/4 where c is the constant defined in A085011.

Links

Table of n, a(n) for n=0..103.

Formula

Product_{k>=0} (1-1/(2^k+1)).

From Robert FERREOL, Feb 28 2020: (Start)

Equals Product_{k>=0} (1 + 1/2^k)^(-1) = 1/A081845.

Equals 1 + Sum_{k>=1} (-1)^k*2^(k*(k+1)/2)/(2-1)*(2^2-1)*...*(2^k-1)). (End)

From Peter Bala, Jan 16 2021: (Start)

Constant C = 2^(-1)*Sum_{n >= 0} (-1/2)^n/Product_{k = 1..n} (1 - 1/2^k).

C = (2^2/(3*5))*Sum_{n >= 0} (-1/8)^n/Product_{k = 1..n} (1 - 1/2^k).

C = (2^9/(3*5*9*17))*Sum_{n >= 0} (-1/32)^n/Product_{k = 1..n} (1 - 1/2^k).

C = (2^20/(3*5*9*17*33*65))*Sum_{n >= 0} (-1/128)^n/Product_{k = 1..n} (1 - 1/2^k) and so on. (End)

Example

0.2097112208975537988549780538514871...

Mathematica

RealDigits[1/QPochhammer[-1, 1/2], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

Prog

(PARI) prod(k=0, 1000, 1-1./(2^k+1))
(PARI) prodinf(k=0, 1-1/(2^k+1)) \\ Michel Marcus, Feb 28 2020

Crossrefs

Cf. A081845, A085011, A261584.

Sequence in context: A152566 A021481 A029686 * A154937 A037996 A299626

Adjacent sequences: A083861 A083862 A083863 * A083865 A083866 A083867

Keyword

nonn,cons

Author

Benoit Cloitre, Jun 19 2003

Status

approved