A262427 -id:A262427 - cp's OEIS frontend

A371824 Decimal expansion of Pi^(1/2)Gamma(1/10)/(5Gamma(3/5)).

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

Offset: 1

Takayuki Tatekawa, Apr 07 2024

Constants from generalized Pi integrals: the case of n=10.

			2.264617395043150744291188990313...

Takayuki Tatekawa, Table of n, a(n) for n = 1..10001

Cf. A085565, A113477, A262427.

RealDigits[2*Sqrt[Pi]/10*Gamma[1/10]/Gamma[3/5], 10, 5001][[1]]
RealDigits[GoldenRatio * Gamma[1/5] * Gamma[2/5]^2 / (2^(6/5) * Sqrt[5] * Pi), 10, 120][[1]] (* Vaclav Kotesovec, Apr 07 2024 *)

Equals 2*Integral_{x=0..1} dx/sqrt(1-x^10).

Equals phi * Gamma(1/5) * Gamma(2/5)^2 / (2^(6/5) * sqrt(5) * Pi), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Apr 07 2024

A371930 Decimal expansion of Pi^(1/2)Gamma(1/14)/(7Gamma(4/7)).

Original entry on oeis.org

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

Offset: 1

Takayuki Tatekawa, Apr 12 2024

Constants from generalized Pi integrals: the case of n=14.

In general, for k > 0, Integral_{x=0..1} 1/sqrt(1 - x^k) dx = 2^(2/k) * Gamma(1 + 1/k)^2 / Gamma(1 + 2/k) = 2^(2/k - 1) * Gamma(1/k)^2 / (k*Gamma(2/k)). - Vaclav Kotesovec, Apr 15 2024

			2.191450245201078533946264870311...

Takayuki Tatekawa, Table of n, a(n) for n = 1..10001

Cf. A085565, A113477, A262427, A371824.

Beta(1/14, 1/2) / 7: evalf(%, 90); # Peter Luschny, Apr 14 2024

RealDigits[Sqrt[Pi]/7*Gamma[1/14]/Gamma[4/7], 10, 5001][[1]]

Equals 2*Integral_{x=0..1} dx/sqrt(1-x^14).
Equals Beta(1/14, 1/2) / 7. - Peter Luschny, Apr 14 2024
Equals Gamma(1/14)^2 / (7 * 2^(6/7) * Gamma(1/7)). - Vaclav Kotesovec, Apr 15 2024

A371929 Decimal expansion of Pi^(1/2)Gamma(1/12)/(6Gamma(7/12)).

Original entry on oeis.org

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

Offset: 1

Takayuki Tatekawa, Apr 12 2024

Constants from generalized Pi integrals: the case of n=12.

			2.2221586039664144669155853439....

Takayuki Tatekawa, Table of n, a(n) for n = 1..10001

Cf. A085565, A113477, A262427, A371824.

Beta(1/12, 1/2) / 6: evalf(%, 89); # Peter Luschny, Apr 14 2024

RealDigits[Sqrt[Pi]/6*Gamma[1/12]/Gamma[7/12], 10, 5001][[1]]
RealDigits[(1 + Sqrt[3]) * Gamma[1/4]^2 / (4 * 3^(3/4) * Sqrt[Pi]), 10, 120][[1]] (* Vaclav Kotesovec, Apr 15 2024 *)

Equals 2*Integral_{x=0..1} dx/sqrt(1-x^12).
Equals Beta(1/12, 1/2) / 6. - Peter Luschny, Apr 14 2024
Equals (1 + sqrt(3)) * Gamma(1/4)^2 / (4 * 3^(3/4) * sqrt(Pi)). - Vaclav Kotesovec, Apr 15 2024

A372327 Decimal expansion of Pi^(1/2)Gamma(1/18)/(9Gamma(5/9)).

Original entry on oeis.org

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

Offset: 1

Takayuki Tatekawa, Apr 28 2024

Constant from generalized Pi integrals: the case of n=18.

			2.14999545849204723391222945664...

Takayuki Tatekawa, Table of n, a(n) for n = 1..10001

Cf. A085565, A113477, A262427, A371929, A371930.

RealDigits[Sqrt[Pi]/9*Gamma[1/18]/Gamma[5/9], 10, 5001][[1]]

Equals 2*Integral_{x=0..1} dx/sqrt(1-x^18).

Equals Gamma(1/18)^2 / (9 * 2^(8/9) * Gamma(1/9)). - Vaclav Kotesovec, Apr 29 2024

A373534 Decimal expansion of Pi^(1/2)Gamma(1/20)/(10Gamma(11/20)).

Original entry on oeis.org

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

Offset: 1

Takayuki Tatekawa, Jun 08 2024

Constants from generalized Pi integrals: the case of n=20.

			2.135344933248004228046475279683...

Takayuki Tatekawa, Table of n, a(n) for n = 1..10001

Cf. A085565, A113477, A262427, A371824, A371881, A371930, A372111, A372327.

(2*sqrt(Pi)*GAMMA(21/20))/GAMMA(11/20): evalf(%, 102); # Peter Luschny, Jun 17 2024

RealDigits[2*Sqrt[Pi]/20*Gamma[1/20]/Gamma[11/20], 10, 5001][[1]]

Equals 2*Integral_{x=0..1} dx/sqrt(1-x^20).

Equals (2*sqrt(Pi)*Gamma(21/20))/Gamma(11/20). - Peter Luschny, Jun 17 2024

A363747 Decimal expansion of 2*Integral_{x=0..1} 1/sqrt(1-x^16) dx.

Original entry on oeis.org

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

Offset: 1

Christian N. Hofmann, Jun 19 2023

Let I(k) = 2*Integral_{x=0..1} 1/sqrt(1-x^(2^k)) dx. Then I(1) = Pi (cf. A000796), I(2) = Gauss lemniscate constant (cf. A062539), I(3) = sqrt(2)*K(sqrt(2)-1) (cf. A262427), I(4) = this constant.

Also I(k) = (Pi^(3/2)*Product_{j=2..k} cos(Pi/(2^j)))/(Gamma(1/2+1/(2^k))*Gamma(1-1/(2^k))). - Christian N. Hofmann, Aug 25 2023

			2.1682048381784119930017239089489332786586588673422...
Gamma(1/16) = 2^(45/16)*Pi^(1/16)*I(2)^(1/8)*I(3)^(1/4)*I(4)^(1/2) = 15.481281...

Cf. A000796, A062539, A262427.

evalf(2*int(1/sqrt(1-t^16),t=0..1), 120);

RealDigits[Beta[1/16, 1/2]/8, 10, 120][[1]] (* Amiram Eldar, Jun 22 2023 *)

2*intnum(x=0, 1, 1/sqrt(1-x^16)) \\ Michel Marcus, Jun 22 2023

Equals Beta(1/16,1/2)/8 = 2*sqrt(Pi)*Gamma(17/16)/Gamma(9/16).
Equals Pi^(3/2)/(8*sin(Pi/16)*Gamma(9/16)*Gamma(15/16)). - Christian N. Hofmann, Aug 28 2023
Gamma(1/2^n) = 2^((n-1)*(1-1/2^n)) * Product_{k=1..n} I(k)^(1/(2^(n-k+1))).

cp's OEIS Frontend

A371824 Decimal expansion of Pi^(1/2)*Gamma(1/10)/(5*Gamma(3/5)).

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A371930 Decimal expansion of Pi^(1/2)*Gamma(1/14)/(7*Gamma(4/7)).

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A371929 Decimal expansion of Pi^(1/2)*Gamma(1/12)/(6*Gamma(7/12)).

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A372327 Decimal expansion of Pi^(1/2)*Gamma(1/18)/(9*Gamma(5/9)).

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A373534 Decimal expansion of Pi^(1/2)*Gamma(1/20)/(10*Gamma(11/20)).

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A363747 Decimal expansion of 2*Integral_{x=0..1} 1/sqrt(1-x^16) dx.

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A371824 Decimal expansion of Pi^(1/2)Gamma(1/10)/(5Gamma(3/5)).

A371930 Decimal expansion of Pi^(1/2)Gamma(1/14)/(7Gamma(4/7)).

A371929 Decimal expansion of Pi^(1/2)Gamma(1/12)/(6Gamma(7/12)).

A372327 Decimal expansion of Pi^(1/2)Gamma(1/18)/(9Gamma(5/9)).

A373534 Decimal expansion of Pi^(1/2)Gamma(1/20)/(10Gamma(11/20)).