A197070 -id:A197070 - cp's OEIS frontend

A350761 Decimal expansion of Pi^2log(2)/6 - log(2)^3/3 - 3zeta(3)/4.

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

Offset: 0

Amiram Eldar, Jan 14 2022

			0.12763051594351388351849171032151833742418129365974...

Ovidiu Furdui, Problem H-761, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 52, No. 4 (2014), p. 374; A Series Whose Sum Involves Pi, ln 2 and zeta(3), Solution to Problem H-761 by AN-anduud Problem Solving Group, ibid., Vol. 54, No. 3 (2016), pp. 283-285.

Cf. A000796, A002117, A002162, A013661, A197070.

RealDigits[Pi^2*Log[2]/6 - Log[2]^3/3 - 3*Zeta[3]/4, 10, 100][[1]]

Equals Sum_{n>=1} ((1/n) * (Sum_{k>=1} (-1)^(k+1)/(n+k))^2).

A354857 Decimal expansion of Sum_{k,m>=1} (-1)^(k+m+1)/floor(sqrt(k+m))^3.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 09 2022

			0.61407153661436281360363608364760166961123536570828...

Ovidiu Furdui, Problem 1849, Mathematics Magazine, Vol. 83, No. 3 (2010), p. 227; An alternating double series, Solution to Problem 1849 by Omran Kouba, ibid., Vol. 84, No. 3 (2010), pp. 234-235.

Cf. A002117, A002162, A072691, A197070.

RealDigits[Pi^2/12 + Log[2] - 3*Zeta[3]/4, 10, 100][[1]]

Equals Pi^2/12 + log(2) - 3*zeta(3)/4.

A373513 Decimal expansion of 3*zeta(3)/2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 07 2024

			1.80308535473939142809960724226717498614747943851...

R. Barbieri, J. A. Mignaco and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, Table II (2).
Chenli Li, Wenchang Chu, Improper integrals involving powers of Inverse Trigonometric and hyperbolic Functions, Mathematics 10 (16) (2022) 2980

Cf. A002117, A197070, A258750.

Maple
```
3*Zeta(3)/2 ; evalf(%) ;
```

RealDigits[3*Zeta[3]/2, 10, 120][[1]] (* Amiram Eldar, Jun 10 2024 *)

3*zeta(3)/2 \\ Michel Marcus, Jun 10 2024

Equals Integral_{x=0..1} log^2(x)/(x+1) dx = -2*Integral_{x=0..1} log(x)*log(1+x)/x dx.
Equals 3*A002117/2 = 2*A197070.
Equals A258750/Pi. - Hugo Pfoertner, Jun 10 2024
Equals Integral_{x=0..1} arctanh^3(x)/x^2 [Li]. - R. J. Mathar, Jun 11 2024

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A350761 Decimal expansion of Pi^2log(2)/6 - log(2)^3/3 - 3zeta(3)/4.

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A354857 Decimal expansion of Sum_{k,m>=1} (-1)^(k+m+1)/floor(sqrt(k+m))^3.

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A373513 Decimal expansion of 3*zeta(3)/2.

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A350761 Decimal expansion of Pi^2*log(2)/6 - log(2)^3/3 - 3*zeta(3)/4.

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A354857 Decimal expansion of Sum_{k,m>=1} (-1)^(k+m+1)/floor(sqrt(k+m))^3.

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A373513 Decimal expansion of 3*zeta(3)/2.

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A350761 Decimal expansion of Pi^2log(2)/6 - log(2)^3/3 - 3zeta(3)/4.