A060804 -id:A060804 - cp's OEIS frontend

A152648 Decimal expansion of 2*zeta(3).

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

Offset: 1

R. J. Mathar, Dec 10 2008

A division by 2 is missing in Mezo's penultimate formula on page 4.

This constant is irrational but not known to be transcendental. - Charles R Greathouse IV, Sep 02 2024

			Equals 2.4041138063191885707994...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Ilham A. Aliev and Ayhan Dil, Tornheim-like series, harmonic numbers and zeta values, arXiv:2008.02488 [math.NT], 2020, p. 2.
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. (3).
David Borwein and J. M. Borwein, On an intriguing integral and some series related to zeta(4), Proc. Am. Math. Soc. 123 (1995) 1191-1198.
Istvan Mezo, Summation of Hyperharmonic Numbers, arXiv:0811.0042 [math.CO], 2008.
Michael Penn, a nice double sum., YouTube video, 2020.
Michael Penn, Euler's harmonic number identity, YouTube video, 2020.

Cf. A002117, A001008, A002805.

Cf. A060804 (continued fraction).

RealDigits[2*Zeta[3],10,120][[1]] (* Harvey P. Dale, Dec 02 2011 *)

default(realprecision, 20080); x=2*zeta(3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b152648.txt", n, " ", d));  \\ Harry J. Smith, Jul 12 2009

Equals 2*A002117 = Sum_{j>=1} H(j)/j^2 where H(j) = A001008(j)/A002805(j).
Equals Integral_{x>=0} x^2/(exp(x)-1). - Jean-François Alcover, Nov 12 2013
Equals Sum_{m>=1} Sum_{n>=1} 1/(m*n*(m + n)). - Jean-François Alcover, Jun 17 2020
Equals Integral_{x=0..1} log(x)^2/(1-x) dx. - Amiram Eldar, Aug 03 2020
Equals the absolute value of psi''(1) = -2.404..., the 2nd derivative of the digamma function at 1. - R. J. Mathar, Aug 29 2023

A060805 Numerators of special continued fraction for 2*zeta(3).

Original entry on oeis.org

2, 1, 2, 1, 4, 2, 6, 4, 9, 6, 12, 9, 16, 12, 20, 16, 25, 20, 30, 25, 36, 30, 42, 36, 49, 42, 56, 49, 64, 56, 72, 64, 81, 72, 90, 81, 100, 90, 110, 100, 121, 110, 132, 121, 144, 132, 156, 144, 169, 156, 182, 169, 196, 182, 210, 196, 225, 210, 240, 225, 256, 240, 272, 256

Offset: 1

N. J. A. Sloane, Apr 29 2001

Y. V. Nesterenko, A few remarks on zeta(3), Mathematical Notes, 59 (No. 6, 1996), 625-636.

Y. V. Nesterenko, Zeta(3) and recurrence relations.
Yu. V. Nesterenko, A few remarks on zeta(3), Math. Notes 59 (1996) 625-636. [From _R. J. Mathar_, Jul 31 2010]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

Cf. A060804, A060806, A060807, A060808, A008733.

Cf. A152648 (2*zeta(3)).

A060805 := proc(n) local nshf,k ; if n <= 2 then op(n,[2,1]) ; else nshf := n-1 ; k := floor(nshf/4) ; if nshf mod 4 = 1 then k*(k+1) ; elif nshf mod 4 = 0 then (k+1)^2 ; elif nshf mod 4 = 2 then (k+1)*(k+2) ; else (k+1)^2 ; end if; end if; end proc: seq(A060805(n),n=1..80) ; # R. J. Mathar, Jul 31 2010

Join[{2, 1}, LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {2, 1, 4, 2, 6, 4, 9}, 100]] (* Jean-François Alcover, Apr 01 2020 *)

a(n) = A008733(n-1), n>2. - R. J. Mathar, Jul 31 2010

More terms from R. J. Mathar, Jul 31 2010

A060806 Denominators of special continued fraction for 2*zeta(3).

Original entry on oeis.org

2, 4, 3, 2, 4, 6, 5, 4, 6, 8, 7, 6, 8, 10, 9, 8, 10, 12, 11, 10, 12, 14, 13, 12, 14, 16, 15, 14, 16, 18, 17, 16, 18, 20, 19, 18, 20, 22, 21, 20, 22, 24, 23, 22, 24, 26, 25, 24, 26, 28, 27, 26, 28, 30, 29, 28, 30, 32, 31, 30, 32, 34, 33, 32, 34, 36, 35, 34, 36

Offset: 1

N. J. A. Sloane, Apr 29 2001

Y. V. Nesterenko, A few remarks on zeta(3), Mathematical Notes, 59 (No. 6, 1996), 625-636.

Y. V. Nesterenko, Zeta(3) and recurrence relations.
Yu. V. Nesterenko, A few remarks on zeta(3), Math. Notes 59 (1996) 625-636.

Cf. A060804-A060808.

a(4*k+1) = 2*k+2, a(4*k+2) = 2*k+4, a(4*k+3) = 2*k+3, a(4*k+4) = 2*k+2 [from Nesterenko]. - Sean A. Irvine, Jan 01 2023

More terms from Sean A. Irvine, Jan 01 2023

A060808 Denominators of ordinary continued fraction convergents for 2*zeta(3).

Original entry on oeis.org

1, 2, 5, 47, 146, 1507, 1653, 8119, 147795, 155914, 615537, 3233599, 10316334, 13549933, 50966133, 166448332, 217414465, 383862797, 2136728450, 6794048147, 56489113626, 63283161773, 183055437172, 246338598945, 15456048571762

Offset: 0

N. J. A. Sloane, Apr 29 2001

			2, 5/2, 12/5, 113/47, 351/146, 3623/1507, 3974/1653, ...

Y. V. Nesterenko, Some remarks on zeta(3), Mathematical Notes, 59 (No. 6, 1996), 625-636.

Y. V. Nesterenko, Zeta(3) and recurrence relations.

Cf. A060804, A060805, A060806, A060807 (numerators).

Digits := 100: t1 := evalf(2*Zeta(3)); cfrac(t1,l1,l2); l1;

More terms from Vladeta Jovovic, Apr 29 2001

Offset changed by Andrew Howroyd, Jul 10 2024

A060807 Numerators of ordinary continued fraction convergents for 2*zeta(3).

Original entry on oeis.org

2, 5, 12, 113, 351, 3623, 3974, 19519, 355316, 374835, 1479821, 7773940, 24801641, 32575581, 122528384, 400160733, 522689117, 922849850, 5136938367, 16333664951, 135806257975, 152139922926, 440086103827, 592226026753

Offset: 0

N. J. A. Sloane, Apr 29 2001

			2, 5/2, 12/5, 113/47, 351/146, 3623/1507, 3974/1653, ...

Y. V. Nesterenko, Some remarks on zeta(3), Mathematical Notes, 59 (No. 6, 1996), 625-636.

Y. Nesterenko, Zeta(3) and recurrence relations.

Cf. A060804, A060805, A060806, A060808 (denominators).

Digits := 100: t1 := evalf(2*Zeta(3)); cfrac(t1,l1,l2); l1;

More terms from Vladeta Jovovic, Apr 29 2001

Offset changed by Andrew Howroyd, Jul 10 2024

cp's OEIS Frontend

A152648 Decimal expansion of 2*zeta(3).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A060805 Numerators of special continued fraction for 2*zeta(3).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A060806 Denominators of special continued fraction for 2*zeta(3).

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A060808 Denominators of ordinary continued fraction convergents for 2*zeta(3).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Extensions

A060807 Numerators of ordinary continued fraction convergents for 2*zeta(3).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Extensions