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 1-3 of 3 results.

A250327 Numerator of the harmonic mean of the first n pentagonal numbers.

Original entry on oeis.org

1, 5, 180, 2640, 23100, 157080, 183260, 2408560, 317026710, 10215305100, 89894684880, 19613385792, 403708857552, 17825298787296, 8681152006800, 435215087274240, 2312080151144400, 43249499297877600, 45652249258870800, 2835244953971976000, 4394629678656562800
Offset: 1

Views

Author

Colin Barker, Nov 18 2014

Keywords

Examples

			a(3) = 180 because the first 3 pentagonal numbers are [1,5,12] and 3/(1/1+1/5+1/12) = 180/77.

Links

Crossrefs

Cf. A250328 (denominators).

Programs

  • Mathematica
    Table[Numerator[n/Total[1/PolygonalNumber[5,Range[n]]]],{n,30}] (* Harvey P. Dale, Jun 01 2022 *)
  • PARI
    harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])
s=vector(30); for(k=1, #s, s[k]=numerator(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

A294513 Denominators of the partial sums of the reciprocals of twice the pentagonal numbers.

Original entry on oeis.org

2, 5, 120, 1320, 9240, 52360, 52360, 602140, 70450380, 2043061020, 16344488160, 3268897632, 62109055008, 2546471255328, 1157486934240, 54401885909280, 272009429546400, 4805499921986400, 4805499921986400, 283524495397197600, 418536159872053600
Offset: 0

Views

Author

Wolfdieter Lang, Nov 02 2017

Keywords

Comments

The corresponding numerators are given by A250328(n+1), n >= 0.
Twice the positive pentagonal numbers are A049450(k+1) = (k+1)*(3*k+2), k >= 0.
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] = [3,2].
The limit of the series is V(3,2) = lim_{n -> oo} V(3,2;n) = (3/2)*log(3) - Pi/(2*sqrt(3)) = 0.74101875088505561179... given in A294514.

Examples

			The rationals V(3,2;n), n >= 0, begin: 1/2, 3/5, 77/120, 877/1320, 6271/9240, 36049/52360, 36423/52360, 422137/602140, 49691099/70450380, 1448086909/2043061020, ...
V(3,2;10^4) = 0.7409854223(Maple, 10 digits) to be compared with 0.7410187513 from V(3,2) given in A294514.
Conjecture tests: a(0) = 2 =  A250327(1)/1, 2* a(1) = 5 = 2*A250327(2)/2 = A250327(2), a(2) = 120 = 2*A250327(2)/3 = 2*180/3, ...

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193 (with v_m(r) = ((m-r)/m)*V(m,r)).

Crossrefs

Cf. A049450, A250327(n+1), A250328(n+1), A294512.

Programs

  • Mathematica
    Denominator@ Accumulate@ Array[1/(2 PolygonalNumber[5, #]) &, 21] (* Michael De Vlieger, Nov 02 2017 *)

Formula

a(n) = denominator(V(3,2;n)) with V(3,2;n) = Sum_{k=0..n} 1/((k + 1)*(3*k + 2)) = Sum_{k=0..n} 1/A049450(k+1) = Sum_{k=0..n} (3/(3*k + 2) - 1/(k+1)).
a(n) = 2*A250327(n+1)/(n+1) [conjecture].
Showing 1-3 of 3 results.

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