cp's OEIS Frontend

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.

Showing 1-2 of 2 results.

A250328 Denominator of the harmonic mean of the first n pentagonal numbers.

Original entry on oeis.org

1, 3, 77, 877, 6271, 36049, 36423, 422137, 49691099, 1448086909, 11631128477, 2334008785, 44471893747, 1827784004699, 832564679309, 39202882860913, 196334425398149, 3473612060358899, 3478128507653999, 205449856947685261, 303604578504856471
Offset: 1

Views

Author

Colin Barker, Nov 18 2014

Keywords

Comments

a(n+1) is, for n >= 0, also the numerator of the partial sums of the reciprocals of twice the pentagonal numbers {A049450(k+1)}A294513(n)%20(assuming%20that%20A250327(n+1)/(n+1)%20=%20A294513(n)/2).%20-%20_Wolfdieter%20Lang">{k>=0} with the denominators given in A294513(n) (assuming that A250327(n+1)/(n+1) = A294513(n)/2). - _Wolfdieter Lang, Nov 02 2017

Examples

			a(3) = 77 because the pentagonal numbers A000326(n), for n = 1,2,3 are 1, 5, 12 and 3/(1/1+1/5+1/12) = 180/77.

Links

Crossrefs

Cf. A000326, A250327 (numerators).

Programs

  • Mathematica
    With[{s = Array[PolygonalNumber[5, #] &, 21]}, Denominator@ Array[HarmonicMean@ Take[s, #] &, Length@ s]] (* Michael De Vlieger, Nov 02 2017 *)
  • PARI
    harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])
s=vector(30); for(k=1, #s, s[k]=denominator(harmonicmean(vector(k, i, (3*i^2-i)/2)))); s

A294514 Decimal expansion of (3/2)*log(3) - Pi/(2*sqrt(3)).

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Nov 02 2017

Keywords

Comments

This is the limit of the series V(3,2) := Sum_{k>=0} 1/((k + 1)*(3*k + 1)) = Sum_{k>=0} 1/A049450(k+1) = (1/2)*Sum_{k>=0} (3/(3*k + 1) - 1/(k+1)) with partial sums given in A250328(n+1)/A294513(n).

Examples

			0.7410187508850556117958287265627106908292027126877538898170990327...

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193, with v_2(3) = (1/3)*V(3,2).

Crossrefs

Cf. A049450, A250328/A294513.

Programs

  • Mathematica
    RealDigits[N[(3/2)*Log[3] - Pi/(2*Sqrt[3]), 157]][[1]] (* Georg Fischer, Apr 04 2020 *)
  • PARI
    (3/2)*log(3) - Pi/(2*sqrt(3)) \\ Michel Marcus, Nov 02 2017

Formula

Equals V(3,2) = Sum_{k>=0} 1/((k + 1)*(3*k + 1)).
Equals Sum_{k>=2} zeta(k)/3^(k-1). - Amiram Eldar, May 31 2021

Extensions

a(100) ff. corrected by Georg Fischer, Apr 04 2020
Data truncated by Sean A. Irvine, Apr 10 2020
Showing 1-2 of 2 results.

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