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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A294965 Denominators of the partial sums of the reciprocals of the numbers (k + 1)*(6*k + 5) = A049452(k+1).

Original entry on oeis.org

5, 110, 5610, 258060, 1496748, 17462060, 715944460, 67298779240, 32101517697480, 378797908830264, 24621864073967160, 1748152349251668360, 1748152349251668360, 145096644987888473880, 2582720280784414835064, 490716853349038818662160, 49562402188252920684878160
Offset: 0

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Author

Wolfdieter Lang, Nov 27 2017

Keywords

Comments

The corresponding numerators are given in A294964. There details are found.

Examples

			For the rationals V(6,5;n) see A294964.

Links

Crossrefs

Cf. A049452, A294964.

Programs

  • Maple
    map(denom,  ListTools:-PartialSums([seq(1/(k+1)/(6*k+5),k=0..20)])); # Robert Israel, Nov 29 2017
  • PARI
    a(n) = denominator(sum(k=0, n, 1/((k + 1)*(6*k + 5)))); \\ Michel Marcus, Nov 27 2017

Formula

a(n) = denominator(V(6,5;n)) with V(6,5;n) = Sum_{k=0..n} 1/((k + 1)*(6*k + 5)) = Sum_{k=0..n} 1/A049452(k+1) = Sum_{k=0..n} (1/(k + 5/6) - 1/(k + 1)).

A294966 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(6*k+5) = A049452(k+1) for k >= 0.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Nov 27 2017

Keywords

Comments

In the Koecher reference v_6(5) = (1/6)*(present value V(6,5)) = 0.05225229129512..., given on p. 192 as (1/4)*log(3) + (1/3)*log(2) - Pi/(4*sqrt(3)).

Examples

			0.313513747770728380036214711836908094696136733315523822488574116360843...

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193.

Links

Crossrefs

Cf. A001620, A049452, A294964/A294965.

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); (3/2)*Log(3) + 2*Log(2) - (1/2)*Pi(R)*Sqrt(3); // G. C. Greubel, Sep 05 2018
  • Mathematica
    RealDigits[-PolyGamma[0, 5/6] + PolyGamma[0, 1], 10, 100][[1]] (* G. C. Greubel, Sep 05 2018 *)
  • PARI
    default(realprecision, 100); (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) \\ G. C. Greubel, Sep 05 2018

Formula

Sum_{k>=0} 1/((6*n + 5)*(n + 1)) =: V(6,5) = (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) = -Psi(5/6) + Psi(1) with the digamma function Psi and Psi(1) = -gamma = A001620.
The partial sums of this series are given in A294964/A294965.
Equals Sum_{k>=2} zeta(k)/6^(k-1). - Amiram Eldar, May 31 2021
Showing 1-2 of 2 results.

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