A389990 - OEIS

%I #33 Oct 30 2025 18:54:11

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

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

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

%N Decimal expansion of the product of alternating ratio's of consecutive positive integer pairs.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lemniscate_constant#Series">Lemniscate constant</a>

%F Equals Product_{x=1..oo} ((2*x-1)/(2*x))^((-1)^(x mod 2)).

%F Equals sqrt(2*Pi)/Gamma(3/4)^2. - _Alois P. Heinz_, Oct 21 2025

%F From _Jwalin Bhatt_, Oct 22 2025: (Start)

%F Equals 2*A062539/Pi = 2*A014549 = 1/A076390.

%F Equals 4/Beta(1/2, 3/4), where Beta(a,b) is the "Beta function" or "Eulerian integral of the first kind".

%F Equals 2/agm(1, sqrt(2)), where agm(y,z) is the Arithmetic-geometric mean. (End)

%e 1.66925368334814637256... = 2/1 * 3/4 * 6/5 * 7/8 * 10/9 * ...

%t RealDigits[Sqrt[2*Pi]/Gamma[3/4]^2, 10, 120][[1]] (* _Amiram Eldar_, Oct 21 2025 *)

%Y Cf. A019727, A068465, A062539, A014549, A076390.

%K nonn,cons,easy

%O 1,2

%A _Marc Morgenegg_, Oct 21 2025

%E More digits from _Alois P. Heinz_, Oct 21 2025