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.

A348131 a(n) is the numerator of the relativistic sum of n velocities of 1/n, in units where the speed of light is 1.

Original entry on oeis.org

1, 4, 7, 272, 211, 51012, 14197, 18640960, 1690981, 11225320100, 313968931, 10079828372880, 83828316391, 12627774819845668, 30436810578889, 21046391759976988928, 14425381885981321, 45032132922921758270916, 8649148282327007911, 120314227994702795221920400
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2021

Keywords

Examples

			The fractions begins with 1, 4/5, 7/9, 272/353, 211/275, 51012/66637, 14197/18571, 18640960/24405761, 1690981/2215269, 11225320100/14712104501, ...
For n = 2, the sum of two velocities of 1/2 is (1/2 + 1/2)/(1 + (1/2)*(1/2)) = 4/5, thus a(2) = 4.

Links

Crossrefs

Cf. A073744, A348051, A348052, A348132 (denominators).

Programs

  • Mathematica
    f[n_] := Module[{s = 1/n}, Do[s = (s + 1/n)/(1 + s/n), {k, 1, n - 1}]; s]; Numerator @ Array[f, 20]

Formula

a(n)/A348132(n) = tanh(n * arctanh(1/n)).
Lim_{n->oo} a(n)/A348132(n) = tanh(1) (A073744).
a(2n-1) = n^(2n-1) - (n-1)^(2n-1) and a(2n) = ((2n+1)^(2n) - (2n-1)^(2n)) / 2. - Thomas Ordowski, Feb 12 2022

A348132 a(n) is the denominator of the relativistic sum of n velocities of 1/n, in units where the speed of light is 1.

Original entry on oeis.org

1, 5, 9, 353, 275, 66637, 18571, 24405761, 2215269, 14712104501, 411625181, 13218256749601, 109949704423, 16565151205544957, 39931933598775, 27614800115689879553, 18928981513351817, 59095217374989483261925, 11350851717672992089, 157904201452248753415276001
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2021

Keywords

Examples

			For n = 2, the sum of two velocities of 1/2 is (1/2 + 1/2)/(1 + (1/2)*(1/2)) = 4/5, thus a(2) = 5.

Links

Crossrefs

Cf. A348051, A348052, A348131 (numerators).

Programs

  • Mathematica
    f[n_] := Module[{s = 1/n}, Do[s = (s + 1/n)/(1 + s/n), {k, 1, n - 1}]; s]; Denominator @ Array[f, 20]

Formula

a(2n-1) = n^(2n-1) + (n-1)^(2n-1) and a(2n) = ((2n+1)^(2n) + (2n-1)^(2n)) / 2. - Thomas Ordowski, Feb 12 2022

A348052 Triangle T(j,k) of denominators of relativistically added fractional velocities w(u,v)=(u+v)/(u*v+1), with velocities enumerated by the Farey series, i.e., u(m) = v(m) = A007305(m)/A007306(m), m>=2.

Original entry on oeis.org

5, 7, 5, 8, 11, 13, 3, 13, 14, 17, 4, 17, 19, 22, 29, 13, 9, 21, 23, 31, 17, 11, 15, 18, 19, 26, 29, 25, 11, 2, 17, 7, 9, 7, 23, 13, 16, 23, 5, 2, 13, 41, 34, 37, 53, 19, 27, 6, 7, 46, 49, 41, 43, 62, 73, 17, 3, 27, 31, 41, 11, 37, 19, 11, 13, 29, 6, 25, 29, 32, 43, 47, 40, 13, 19, 68, 61, 65
Offset: 2

Views

Author

Hugo Pfoertner, Sep 25 2021

Keywords

Examples

			See A348051.
The triangle starts:
  4/5,
  5/7,  3/5,
  7/8,  9/11, 12/13,
  2/3,  7/13, 11/14,  8/17,
  3/4, 11/17, 16/19, 13/22, 20/29
....

Crossrefs

A348051 are the corresponding numerators.
Cf. A007305, A007306, A348131, A348132.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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