A378130 -id:A378130 - cp's OEIS frontend

A378128 Decimal expansion of 2/L, where L is the lemniscate constant (A062539).

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

Offset: 0

Paolo Xausa, Nov 17 2024

			0.76275976350181318806232598096361579329262923734807...

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

Cf. A010466, A062539, A085565, A175575.

Cf. A377999, A378129, A378130, A378131, A378132.

First[RealDigits[Sqrt[8]*Gamma[3/4]^2/Pi^(3/2), 10, 100]]

Equals 1/A085565.
Equals 2*sqrt(2)*Gamma(3/4)^2/Pi^(3/2) = A010466*A175575.
Equals Product_{k >= 1} b(k), where b(1) = sqrt(1/2) and, for k >= 2, b(k) = sqrt(1/2 + (1/2)/b(k-1)).

A378129 Decimal expansion of log(L^2/Pi), where L is the lemniscate constant (A062539).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 17 2024

			0.78318878541367355294389069379822205618042023154005...

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

Cf. A002388, A062539, A068465, A371855.

Cf. A377999, A378128, A378130, A378131, A378132.

First[RealDigits[Log[Pi^2/2/Gamma[3/4]^4], 10, 100]] (* or *)
First[RealDigits[Integrate[Exp[-x]*Tanh[x]/x, {x, 0, Infinity}], 10, 100]]

Equals log(Pi^2/(2*Gamma(3/4)^4)) = log(A002388/(2*A068465^4)).
Equals 2*log(2*Gamma(5/4)/Gamma(3/4)) = 2*log(A371855/A068465).
Equals Integral_{x=0..oo} exp(-x)*tanh(x)/x dx.

A378131 Decimal expansion of sqrt(1 + sqrt(3))L/(Pi12^(1/8)), where L is the lemniscate constant (A062539).

Original entry on oeis.org

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

Paolo Xausa, Nov 18 2024

			1.011204695537690090572855988569625803283536658...

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

Cf. A000796, A062539, A068465, A090388.

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

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]]

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.

A378132 Decimal expansion of L^4/15, where L is the lemniscate constant (A062539).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 18 2024

			3.15121200215389753821768994224868855664551935451...

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

Cf. A062539, A068465, A092732.

Cf. A377999, A378128, A378129, A378130, A378131.

First[RealDigits[Pi^6/(60*Gamma[3/4]^8), 10, 100]]

Equals Pi^6/(60*Gamma(3/4)^8) = A092732/(60*A068465^8).
Equals Sum_{z is a Gaussian integer != 0} 1/z^4.
Equals G_4(i), where G_4 is the Eisenstein series of weight 4.

cp's OEIS Frontend

A378128 Decimal expansion of 2/L, where L is the lemniscate constant (A062539).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A378129 Decimal expansion of log(L^2/Pi), where L is the lemniscate constant (A062539).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A378132 Decimal expansion of L^4/15, where L is the lemniscate constant (A062539).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A378131 Decimal expansion of sqrt(1 + sqrt(3))L/(Pi12^(1/8)), where L is the lemniscate constant (A062539).