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.

A244647 Decimal expansion of the sum of the reciprocals of the decagonal numbers (A001107).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Jul 03 2014

Keywords

Comments

For the partial sums of the reciprocals of the (positive) decagonal numbers see A250551(n+1)/A294515(n), n >= 0. - Wolfdieter Lang, Nov 07 2017

Examples

			1.216745956158244182494339352004760382108361700922772890949837441544696356350....

Links

Crossrefs

Cf. A001107, A000038, A013661, A244639, A016627, A244645, A244646, A244648, A244649, A250551(n+1)/A294515(n).

Programs

  • Mathematica
    RealDigits[ Log[2] + Pi/6, 10, 111][[1]] (* or *)
RealDigits[ Sum[1/(4n^2 - 3n), {n, 1 , Infinity}], 10, 111][[1]]
  • PARI
    log(2)+Pi/6 \\ Charles R Greathouse IV, Feb 08 2023

Formula

Sum_{n>0} 1/(4n^2 - 3n) = log(2) + Pi/6, (A002162 + A019673).

A250551 Denominator of the harmonic mean of the first n positive 10-gonal numbers.

Original entry on oeis.org

1, 11, 307, 8117, 139393, 982381, 4935773, 287319059, 1056494083, 39179109811, 1609331378051, 4835480422963, 33892787092141, 1798339013862173, 34201770221163407, 4176177999344899729, 4179324192635626369, 32062945622467289429, 2341997846273161559117
Offset: 1

Views

Author

Colin Barker, Nov 25 2014

Keywords

Comments

a(n+1) is, for n >= 0, also the numerator of the partial sums of the reciprocal of the positive decagonal numbers A001107(n+1) with the denominators A294515(n) (provided A294515(n) = A250550(n+1)/(n+1)). - Wolfdieter Lang, Nov 02 2017

Examples

			a(3) = 307 because the first 3 positive decagonal numbers A001107 are [1,10,27], and 3/(1/1+1/10+1/27) = 810/307.

Links

Crossrefs

Cf. A001107 (10-gonal numbers), A250550 (numerators).

Programs

  • Mathematica
    With[{s = Array[PolygonalNumber[10, #] &, 19]}, 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(n=1, #s, s[n]=denominator(harmonicmean(vector(n, k, (8*k^2-6*k)/2)))); s

Formula

a(n) = denominator(r(n)) with the rationals r(n) = n/Sum_{k=1..n} A001107(n), n >= 1. See the name. - Wolfdieter Lang, Nov 02 2017

A294516 Numerators of the partial sums of the reciprocals of (k+1)*(4*k+3) = A033991(k+1), for k >= 0.

Original entry on oeis.org

1, 17, 67, 2087, 40577, 315967, 8627249, 539432053, 543008461, 7096662277, 306487877071, 14457409539227, 246534893826499, 49437672710843, 14617658229054773, 29294219493288391, 1966205309547985477, 139821581165897995307, 700098935135639210887, 55378426713778630607653, 4601722042202662057443599, 12144567347216934480292961
Offset: 0

Views

Author

Wolfdieter Lang, Nov 07 2017

Keywords

Comments

The corresponding numerators are given in A294517.
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] = [4,3].
The limit of the series is V(4,3) = lim_{n -> oo} V(4,3;n) = 3*log(2) - Pi/2 = 0.50864521488493930902... given in A294518.

Examples

			The rationals V(4,3;n), n >= 0, begin: 1/3, 17/42, 67/154, 2087/4620, 40577/87780, 315967/672980, 8627249/18170460, 539432053/1126568520, 543008461/1126568520, 7096662277/14645390760, 306487877071/629751802680, ...
V(4,3;10^4) = 0.508620219 (Maple, 10 digits) to be compared with 0.508645215 from V(4,3) given in A294518.

References

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

Links

Crossrefs

Cf. A294512, A250551(n+1)/A294515(n) (V(4,1;n)), A294517, A294518.

Programs

  • PARI
    a(n) = numerator(sum(k=0, n, 1/((k + 1)*(4*k + 3)))); \\ Michel Marcus, Nov 15 2017

Formula

a(n) = numerator(V(4,3;n)) with V(4,3;n) = Sum_{k=0..n} 1/((k + 1)*(4*k + 3)) = Sum_{k=0..n} 1/A033991(k+1) = Sum_{k=0..n} (4/(4*k + 3) - 1/(k+1)).
V(4,3;n) = 3*log(2) - Pi/2 + Psi(n+7/4) - Psi(n+2) with the digamma function Psi. Note that Psi(1) - Psi(3/4) = 3*log(2) - Pi/2. - Wolfdieter Lang, Nov 15 2017
Showing 1-3 of 3 results.

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