A378131 Decimal expansion of sqrt(1 + sqrt(3))*L/(Pi*12^(1/8)), where L is the lemniscate constant (A062539).

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

Offset: 1

Author

Paolo Xausa, Nov 18 2024

Keywords

• nonn
• cons
• easy

Examples

			1.011204695537690090572855988569625803283536658...
		

Links

• Wikipedia, Lemniscate constant (includes this constant).
• Index to sequences related to the Gamma function.

Crossrefs

Cf. A000796, A062539, A068465, A090388.

Cf. A377999, A378128, A378129, A378130, A378132.

Programs

Mathematica

First[RealDigits[Sqrt[(1 + Sqrt[3])*Pi]/(2^(3/4)*3^(1/8)*Gamma[3/4]^2), 10, 100]] (* or *)
First[RealDigits[Hypergeometric2F1[1/3, 2/3, 1, (3*Sqrt[3] - 5)/4], 10, 100]]

Formula

Equals sqrt((1 + sqrt(3))*Pi)/(2^(3/4)*3^(1/8)*Gamma(3/4)^2) = sqrt(A090388*A000796)/(2^(3/4)*3^(1/8)*A068465^2).

Equals Sum_{j,k integers} exp(-2*Pi*(j^2 + j*k + k^2)).

Equals 2F1(1/3, 2/3, 1, (3*sqrt(3) - 5)/4), where 2F1 is the ordinary hypergeometric function.