A378131 -id:A378131 - 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.

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.

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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A378129 Decimal expansion of log(L^2/Pi), 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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A378132 Decimal expansion of L^4/15, where L is the lemniscate constant (A062539).

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