A348051 Triangle T(j,k) of numerators 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.

4, 5, 3, 7, 9, 12, 2, 7, 11, 8, 3, 11, 16, 13, 20, 11, 7, 19, 17, 25, 15, 10, 13, 17, 16, 23, 27, 24, 7, 1, 13, 3, 5, 5, 19, 5, 11, 13, 4, 1, 8, 31, 29, 17, 28, 14, 17, 5, 4, 31, 39, 36, 23, 37, 48, 13, 2, 23, 19, 29, 9, 33, 11, 7, 9, 21, 5, 19, 26, 23, 34, 41, 37, 9, 14, 53, 49, 56

Offset

2,1

Comments

The velocities are assumed to be given in units of c, and thus c = 1.

Links

Table of n, a(n) for n=2..79.

Wikipedia, Velocity-addition formula.

Example

The triangle of added fractions begins:
     u   1/2   1/3   2/3   1/4   2/5   3/5   3/4   1/5   2/7   3/8   3/7
   v \    .     .     .     .     .     .     .     .     .     .     .
  1/2 |  4/5    .     .     .     .     .     .     .     .     .     .
  1/3 |  5/7   3/5    .     .     .     .     .     .     .     .     .
  2/3 |  7/8   9/11 12/13   .     .     .     .     .     .     .     .
  1/4 |  2/3   7/13 11/14  8/17   .     .     .     .     .     .     .
  2/5 |  3/4  11/17 16/19 13/22 20/29   .     .     .     .     .     .
  3/5 | 11/13  7/9  19/21 17/23 25/31 15/17   .     .     .     .     .
  3/4 | 10/11 13/15 17/18 16/19 23/26 27/29 24/25   .     .     .     .
  1/5 |  7/11  1/2  13/17  3/7   5/9   5/7  19/23  5/13   .     .     .
  2/7 | 11/16 13/23  4/5   1/2   8/13 31/41 29/34 17/37 28/53   .     .
  3/8 | 14/19 17/27  5/6   4/7  31/46 39/49 36/41 23/43 37/62 48/73   .
  3/7 | 13/17  2/3  23/27 19/31 29/41  9/11 33/37 11/19  7/11  9/13 21/29

Crossrefs

A348052 are the corresponding denominators.

Cf. A007305, A007306, A348131, A348132.

Sequence in context: A268741 A263031 A004493 * A328238 A170929 A245085

Adjacent sequences: A348048 A348049 A348050 * A348052 A348053 A348054

Keyword

nonn,frac,tabl

Author

Hugo Pfoertner, Sep 25 2021

Status

approved