A378129 -id:A378129 - 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)).

A378130 Decimal expansion of 24L^2/(5^(7/4)Pi^2), where L is the lemniscate constant (A062539).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 18 2024

			0.999996383159084127772763499184706112808943488770...

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

Cf. A008977, A062539, A091670.

Cf. A377999, A378128, A378129, A378129, A378131, A378132.

First[RealDigits[12*Pi/(5^(7/4)*Gamma[3/4]^4), 10, 100]] (* or *)
First[RealDigits[Sum[((-1)^k/6635520^k)*(4*k)!/k!^4, {k, 0, Infinity}], 10, 100]] (* or *)
First[RealDigits[HypergeometricPFQ[{1/4, 1/2, 3/4}, {1, 1}, -1/25920], 10, 100]]

Equals 12*Pi/(5^(7/4)*Gamma(3/4)^4) = 12*A091670/5^(7/4).
Equals Sum_{k >= 0} ((-1)^k/6635520^k)*(4*k)!/(k!)^4 = Sum_{k >= 0} ((-1)^k/6635520^k)*A008977(k).
Equals pFq(1/4, 1/2, 3/4; 1, 1; -1/25920), where pFq is the generalized hypergeometric function.

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).

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A378130 Decimal expansion of 24*L^2/(5^(7/4)*Pi^2), where L is the lemniscate constant (A062539).

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

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

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Formula

A378130 Decimal expansion of 24L^2/(5^(7/4)Pi^2), where L is the lemniscate constant (A062539).

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