A218505 -id:A218505 - cp's OEIS frontend

A238181 Decimal expansion of sum_(n>=1) H(n)^2/n^3 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,3)).

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.6519427927044986239626937611144940161...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 24.

Cf. A152648, A152649, A152651, A218505, A238168, A238182, A238183.

7/2*Zeta[5] - Zeta[2]*Zeta[3] // RealDigits[#, 10, 100]& // First

7/2*zeta(5) - zeta(2)*zeta(3) \\ Stefano Spezia, May 22 2025

7/2*zeta(5) - zeta(2)*zeta(3).

A238182 Decimal expansion of Sum_{n>=1} H(n)^2/n^4 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,4)).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

No closed form of S(2,2q) is known to date, except for S(2,2) (A218505) and S(2,4) (this sequence).

			1.221879945319880138518806475290994812569...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 24.

Cf. A001008/A002805, A152648, A152649, A152651, A218505, A238168, A238181, A238183.

97/24*Zeta[6] - 2*Zeta[3]^2 // RealDigits[#, 10, 100]& // First

97/24*zeta(6) - 2*zeta(3)^2.

A345203 Decimal expansion of zeta(2) + 2*zeta(3).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 10 2021

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Springer, 2013, section 3.71, p. 150.

			4.04904787316741500727189148966892517074892248588779...

Ovidiu Furdui, Series Involving Products of Two Harmonic Numbers, Mathematics Magazine, Vol. 84, No. 5 (2011), pp. 371-377.

Cf. A001008, A002117, A002805, A013661, A218505, A345204.

RealDigits[Zeta[2] + 2*Zeta[3], 10, 100][[1]]

Equals A013661 + 2 * A002117.
Equals Sum_{k>=1} (k+2)/k^3.
Equals Sum_{k>=1} H(k)*H(k+1)/(k*(k+1)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Furdui, 2011).
Equals Sum_{k>=1} (H(k)+1)/k^2.
Equals 1 + Sum_{k>=2} H(k)/(k-1)^2.
Equals Sum_{k>=2} (k-1)^2*(zeta(k)-1).
Equals 3 + Sum_{k>=3} (-1)^(k+1)*k^2*(zeta(k)-1).
Equals Integral_{x=0..1} log(x)*(log(x)-1)/(1-x) dx.
Equals Integral_{x>=1} log(x)*(log(x)+1)/(x*(x-1)) dx.
Equals Integral_{x>=0} x*(x+1)/(exp(x)-1) dx.

A241753 Decimal expansion of Sum_{n>=1} (H(n)/(n+1))^2, where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 28 2014

			2.97638889270563002666901016548821173263...

David H. Bailey and Jonathan M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the AMS Volume 52, Number 5, page 506.

Cf. A218505.

RealDigits[11*Pi^4/360, 10, 100] // First

11*Pi^4/360 \\ Stefano Spezia, May 26 2025

Equals 1/(2*Pi)*Integral_{x=0..Pi} (Pi-t)^2*log(2*sin(t/2))^2 dt.
Equals 11/17*A218505.
Equals 11*Pi^4/360. - Vaclav Kotesovec, Apr 28 2014

A238183 Decimal expansion of sum_(n>=1) H(n)^2/n^7 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,7)).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.019483497494382286206496675928126515...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 24.

Cf. A152648, A152649, A152651, A218505, A238168, A238181, A238182.

Zeta[3]^3/3 - 5/2*Zeta[4]*Zeta[5] - 7/2*Zeta[3]*Zeta[6] - Zeta[2]*Zeta[7] + 55/6*Zeta[9] // RealDigits[#, 10, 100]& // First

zeta(3)^3/3-5/2*zeta(4)*zeta(5)-7/2*zeta(3)*zeta(6)-zeta(2)*zeta(7)+55/6*zeta(9).

A345204 Decimal expansion of 5/4 + Pi^2/8 + zeta(3).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 10 2021

			3.68575745329576411275404953649596888267919871824559...

Ovidiu Furdui, Series Involving Products of Two Harmonic Numbers, Mathematics Magazine, Vol. 84, No. 5 (2011), pp. 371-377.

Cf. A001008, A002117, A002805, A111003, A218505, A345203.

RealDigits[5/4 + Pi^2/8 + Zeta[3], 10, 100][[1]]

5/4 + Pi^2/8 + zeta(3) \\ Stefano Spezia, Jan 30 2025

Equals Sum_{k>=1} H(k)*H(k+2)/(k*(k+2)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Furdui, 2011).

Equals 5/4 + A111003 + A002117.

A238135 Decimal expansion of Euler's Multi-Zeta Sum S(2,3) = Sum_{n >= 1} (Sum_{k=1..n}((-1)^(k + 1)/k)^2/(n + 1)^3).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Feb 18 2014

			0.1561669333811769158810359096879881936857767...

D. H. Bailey and J. M. Borwein, Euler's Multi-Zeta Sums, Lawrence Berkeley National Laboratory, 1995, p. 10.

Cf. A218505.

4*PolyLog[5, 1/2] - 1/30*Log[2]^5 - 17/32*Zeta[5] - 11/720*Pi^4*Log[2] + 7/4*Zeta[3]*Log[2]^2 + 1/18*Pi^2*Log[2]^3 - 1/8*Pi^2*Zeta[3] // RealDigits[#, 10, 100]& // First

4*polylog(5,1/2)-1/30*log(2)^5-17/32*zeta(5) - 11/720*Pi^4*log(2) + 7/4*zeta(3)*log(2)^2 + 1/18*Pi^2*log(2)^3 - 1/8*Pi^2*zeta(3) \\ Charles R Greathouse IV, Jul 18 2014

RR = RealBallField(380)
f = fast_callable(4*polylog(5, 1/2) - 1/30*log(2)^5 - 17/32*zeta(5) - 11/720*pi^4*log(2) + 7/4*zeta(3)*log(2)^2 + 1/18*pi^2*log(2)^3 - 1/8*pi^2*zeta(3), vars=[x], domain=RR)
print([int(t) for t in str(f(0))[3:103]]) # Peter Luschny, May 06 2020

A256593 Decimal expansion of 1/PiIntegral_{0..Pi} x^2log(2*cos(x/2))^2 dx, one of the log-cosine integrals related to zeta(4).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 03 2015

			5.952777785411260053338020330976423465261130235552997992256369...

David Borwein and Jonathan M. Borwein, On an Intriguing Integral and Some Series Related to Zeta(4), Proc. Amer. Math. Soc. 123 (1995), 1191-1198

Cf. A013662, A218505, A256568.

RealDigits[11*Pi^4/180, 10, 102] // First

1/Pi*Integral_{0..Pi} x^2*log(2*cos(x/2))^2 dx = 11*Pi^4/180 = 11/2*zeta(4).

A384457 Decimal expansion of Sum_{k>=1} H(k)^3/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 30 2025

			3.59342794177494296025518240703339219591695480351933...

K. Ramachandra and R. Sitaramachandrarao, On series, integrals and continued fractions - II, Madras Univ. J., Sect. B, 51 (1988), pp. 181-198.

K. Ramachandra, On series integrals and continued fractions I, Hardy-Ramanujan Journal, Vol. 4 (1981), pp. 1-11.
K. Ramachandra, On series, integrals and continued fractions, III, Acta Arithmetica, Vol. 99, No. 3 (2001), pp. 257-266.

Cf. A001008, A002805.
Cf. A000796, A002117, A002162, A352769.
Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.

RealDigits[Zeta[3] + (Pi^2*Log[2] + Log[2]^3)/3, 10, 120][[1]]

PARI
```
zeta(3) + (Pi^2*log(2) + log(2)^3)/3
```

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

A384458 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*H(k)^3/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 30 2025

			0.27412574654925297067883303678750470762654489295575...

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 245, eq. (4.149).
K. Ramachandra and R. Sitaramachandrarao, On series, integrals and continued fractions - II, Madras Univ. J., Sect. B, 51 (1988), pp. 181-198.

K. Ramachandra, On series integrals and continued fractions I, Hardy-Ramanujan Journal, Vol. 4 (1981), pp. 1-11.
K. Ramachandra, On series, integrals and continued fractions, III, Acta Arithmetica, Vol. 99, No. 3 (2001), pp. 257-266.

Cf. A001008, A002805.
Cf. A000796, A002117, A002162, A013662.
Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.

RealDigits[(Pi*Log[2])^2/8 + 5*Zeta[4]/8 - 9*Zeta[3]*Log[2]/8 - Log[2]^4/4, 10, 120][[1]]

(Pi*log(2))^2/8 + 5*zeta(4)/8 - 9*zeta(3)*log(2)/8 - log(2)^4/4

Equals (Pi*log(2))^2/8 + 5*zeta(4)/8 - 9*zeta(3)*log(2)/8 - log(2)^4/4.

cp's OEIS Frontend

A238181 Decimal expansion of sum_(n>=1) H(n)^2/n^3 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,3)).

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A238182 Decimal expansion of Sum_{n>=1} H(n)^2/n^4 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,4)).

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A345203 Decimal expansion of zeta(2) + 2*zeta(3).

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A241753 Decimal expansion of Sum_{n>=1} (H(n)/(n+1))^2, where H(n) is the n-th harmonic number.

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A238183 Decimal expansion of sum_(n>=1) H(n)^2/n^7 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,7)).

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A345204 Decimal expansion of 5/4 + Pi^2/8 + zeta(3).

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A238135 Decimal expansion of Euler's Multi-Zeta Sum S(2,3) = Sum_{n >= 1} (Sum_{k=1..n}((-1)^(k + 1)/k)^2/(n + 1)^3).

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A256593 Decimal expansion of 1/Pi*Integral_{0..Pi} x^2*log(2*cos(x/2))^2 dx, one of the log-cosine integrals related to zeta(4).

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A256593 Decimal expansion of 1/PiIntegral_{0..Pi} x^2log(2*cos(x/2))^2 dx, one of the log-cosine integrals related to zeta(4).