cp's OEIS Frontend

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

%I A083864 #35 Jan 11 2025 20:01:40
%S A083864 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,
%T A083864 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,
%U A083864 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
%N A083864 Decimal expansion of Product_{k>=0} (1 - 1/(2^k+1)).
%C A083864 c/4 where c is the constant defined in A085011.
%F A083864 Product_{k>=0} (1-1/(2^k+1)).
%F A083864 From _Robert FERREOL_, Feb 28 2020: (Start)
%F A083864 Equals Product_{k>=0} (1 + 1/2^k)^(-1) = 1/A081845.
%F A083864 Equals 1 + Sum_{k>=1} (-1)^k*2^(k*(k+1)/2)/((2-1)*(2^2-1)*...*(2^k-1)). (End)
%F A083864 From _Peter Bala_, Jan 16 2021: (Start)
%F A083864 Constant C = 2^(-1)*Sum_{n >= 0} (-1/2)^n/Product_{k = 1..n} (1 - 1/2^k).
%F A083864 C = (2^2/(3*5))*Sum_{n >= 0} (-1/8)^n/Product_{k = 1..n} (1 - 1/2^k).
%F A083864 C = (2^9/(3*5*9*17))*Sum_{n >= 0} (-1/32)^n/Product_{k = 1..n} (1 - 1/2^k).
%F A083864 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)
%e A083864 0.2097112208975537988549780538514871...
%t A083864 RealDigits[1/QPochhammer[-1, 1/2], 10, 120][[1]] (* _Amiram Eldar_, May 29 2023 *)
%o A083864 (PARI) prod(k=0,1000,1-1./(2^k+1))
%o A083864 (PARI) prodinf(k=0, 1-1/(2^k+1)) \\ _Michel Marcus_, Feb 28 2020
%Y A083864 Cf. A081845, A085011, A261584.
%K A083864 nonn,cons
%O A083864 0,1
%A A083864 _Benoit Cloitre_, Jun 19 2003