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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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A294831 Numerators of the partial sums of the reciprocals of the numbers (k + 1)*(5*k + 4) = 2*A005476(k+1), for k >= 0.

Original entry on oeis.org

1, 11, 83, 410, 16799, 495151, 8516747, 55850623, 309309419, 1088610631, 6561497681, 777210281963, 12475578306953, 287734917200239, 10671842976127147, 844855135994501953, 846430303832665873, 75457260356268267017, 3551759427031132995079, 711302288219532928235, 712163917143684270659
Offset: 0

Views

Author

Wolfdieter Lang, Nov 18 2017

Keywords

Comments

The corresponding denominators are given in A294832.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [5,4].
The limit of the series is V(5,4) = lim_{n -> oo} V(5,4;n) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) - (Pi/5)*sqrt(3 + 4*phi)))/2, with the golden section phi:= (1 + sqrt(5))/2 = A001622. The value is 0.3877929018046... given in A294833.
In the Koecher reference v_5(4) = (1/5)*V(5,4) = 0.07755858036... given there by (1/4)*log(5) + (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) - (Pi/10)*sqrt((5 + 2*sqrt(5))/5).

Examples

			The rationals V(5,4;n), n >= 0, begin:1/4, 11/36, 83/252, 410/1197, 16799/47880, 495151/1388520, 8516747/23604840, 55850623/153431460, 309309419/843873030, 1088610631/2953555605, 6561497681/17721333630, 777210281963/2091117368340, 12475578306953/33457877893440, ...
V(5,4;10^6) = 0.3877927018 (Maple 10 digits) to be compared with 0.3877929018 obtained from A294833 with 10 digits.

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193. For v_5(4) see p. 192.

Links

Crossrefs

Cf. A001622, 2*A005476, A294512, A294828/A294829 (V(5,3;n)), A294832, A294833.

Programs

  • Magma
    [Numerator((&+[1/((k + 1)*(5*k + 4)): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Aug 30 2018
  • Mathematica
    Table[Numerator[Sum[1/((k + 1)*(5*k + 4)), {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Aug 30 2018 *)
  • PARI
    a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 4)))); \\ Michel Marcus, Nov 19 2017

Formula

a(n) = numerator(V(5,4;n)) with V(5,4;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 4)) = Sum_{k=0..n} 1/(2*A005476(k+1)) = Sum_{k=0..n} (1/(k + 4/5) - 1/(k+1)) = -Psi(4/5) + Psi(n+9/5) - (gamma + Psi(n+2)) with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

A294833 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(5*k+4) = 2*A005476(k+1) for k >= 0.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Nov 18 2017

Keywords

Comments

In the Koecher reference v_5(4) = (1/5)*(present value V(5,4)) = 0.07755858036092111..., given there as (1/4)*log(5) + (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) - (Pi/10)*sqrt((5 + 2*sqrt(5))/5).

Examples

			0.387792901804605598784785543074432988592001153755299230304043559360093480608...

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193. For v_5(4) see p. 192.

Links

Crossrefs

Cf. A001620, A001622, 2*A005476, A294831/A294832.

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); ((5/2)*Log(5) + Sqrt(5)*(Log((1+Sqrt(5))/2) - (Pi(R)/5)*Sqrt(5+2*Sqrt(5))))/2; // G. C. Greubel, Sep 05 2018
  • Mathematica
    RealDigits[-PolyGamma[0, 4/5] + PolyGamma[0, 1], 10, 100][[1]] (* G. C. Greubel, Sep 05 2018 *)
  • PARI
    default(realprecision, 100); phi=(1+sqrt(5))/2; ((5/2)*log(5) + (2*phi - 1)*(log(phi) - (Pi/5)*sqrt(3 + 4*phi)))/2 \\ G. C. Greubel, Sep 05 2018

Formula

Sum_{k>=0} 1/((5*n + 4)*(n + 1)) =: V(5,4) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) - (Pi/5)*sqrt(3 + 4*phi)))/2 = -Psi(4/5) + Psi(1) with the golden section phi =(1 + sqrt(5))/2 = A001622 with the digamma function Psi and Psi(1) = -gamma = A001620.
The partial sums of this series are given in A294831/A294832.
Equals Sum_{k>=2} zeta(k)/5^(k-1). - Amiram Eldar, May 31 2021
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