A389990 Decimal expansion of the product of alternating ratio's of consecutive positive integer pairs.

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, 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, 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

Offset

1,2

Links

Table of n, a(n) for n=1..103.

Wikipedia, Lemniscate constant

Formula

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

Equals sqrt(2*Pi)/Gamma(3/4)^2. - Alois P. Heinz, Oct 21 2025

From Jwalin Bhatt, Oct 22 2025: (Start)

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

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

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

Example

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

Mathematica

RealDigits[Sqrt[2*Pi]/Gamma[3/4]^2, 10, 120][[1]] (* Amiram Eldar, Oct 21 2025 *)

Crossrefs

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

Sequence in context: A019851 A155880 A393648 * A217852 A021603 A250721

Adjacent sequences: A389987 A389988 A389989 * A389991 A389992 A389993

Keyword

nonn,cons,easy

Author

Marc Morgenegg, Oct 21 2025

Extensions

More digits from Alois P. Heinz, Oct 21 2025

Status

approved