A242822 -id:A242822 - cp's OEIS frontend

A377753 Decimal expansion of 8*G/Pi^2, where G is the Catalan's constant (A006752).

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

Offset: 0

Stefano Spezia, Nov 07 2024

			0.7424537454215443259079279607988799424377218365...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.7 and 7.7, pp. 54, 474.

Cf. A002388, A006752, A218387, A242822 (see the second formula).

Mathematica
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RealDigits[8Catalan/Pi^2,10,100][[1]]
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8*Catalan/Pi^2 \\ Charles R Greathouse IV, Nov 27 2024

Equals (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)^2) / (Sum_{n>=1} 1/(2*n - 1)^2) (see Finch).
Equals 1/A242822. - Hugo Pfoertner, Nov 07 2024
Equals (1 - W)/(1 + W), where W = tanh(Sum_{prime p == 3 (mod 4)} arctanh(1/p^2)) = zeta(2,3/4)/zeta(2,1/4) = (Pi^2 - 8*G)/(Pi^2 + 8*G) = 0.1478066521164... Physical interpretation: the constant W is the relativistic sum of the velocities c/p^2 over all primes p == 3 (mod 4), in units where the speed of light c = 1. - Thomas Ordowski, Nov 23 2024

A263809 Decimal expansion of C_{1/2}, a constant related to Kolmogorov's inequalities.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 27 2015

			2.78640785937135371836849252065073648531496243503123575794856326376...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 7.7 Riesz-Kolmogorov Constants, p. 474.

Burgess Davis, On Kolmogorov's Inequalities, Transactions of the American Mathematical Society Vol. 222 (Sep., 1976), pp. 179-192.

Cf. A068465, A068466, A093341, A242822.

RealDigits[Gamma[1/4]^2/(Pi*Gamma[3/4]^2), 10, 105] // First

gamma(1/4)^2/(Pi*gamma(3/4)^2) \\ Michel Marcus, Oct 27 2015

C_{1/2} = gamma(1/4)^2/(Pi*gamma(3/4)^2).
Equals (1/Pi^2)*(integral_{0..Pi} sqrt(csc(t)) dt)^2.
Also equals (8/Pi^2)*A093341^2.

A330890 Decimal expansion of Product_{prime p == 1 (mod 4)} (1 + 1/p^2)/(1 - 1/p^2).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Apr 30 2020

			1.1136806181323164888618919411983199136565827547877592324456...

Cf. A002144, A088539, A242822, A243380, A242822 (see the second formula).

Cf. A334424/A334425, A334445/A334446, A334449/A334450.

RealDigits[12*Catalan/Pi^2, 10, 120][[1]]

12*Catalan/Pi^2 \\ Michel Marcus, May 01 2020

Equals 12*G/Pi^2, where G is Catalan's constant (A006752).
Equals A243380 / A088539.
Equals Sum_{q in A004613} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021
Equals (1 + w)/(1 - w), where w = tanh(Sum_{prime p == 1 (mod 4)} arctanh(1/p^2)) = 0.0537832523783875... Physical interpretation: the constant w is the relativistic sum of the velocities c/p^2 over all Pythagorean primes p, in units where the speed of light c = 1. - Thomas Ordowski, Nov 14 2024

Name edited by Thomas Ordowski, Nov 15 2024

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A377753 Decimal expansion of 8*G/Pi^2, where G is the Catalan's constant (A006752).

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A263809 Decimal expansion of C_{1/2}, a constant related to Kolmogorov's inequalities.

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A330890 Decimal expansion of Product_{prime p == 1 (mod 4)} (1 + 1/p^2)/(1 - 1/p^2).

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