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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.
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%I #7 Oct 07 2021 07:10:24

%S 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,

%T 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,

%U 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

%N 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.

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Velocity-addition_formula">Velocity-addition formula</a>.

%e The triangle of added fractions begins:

%e u 1/2 1/3 2/3 1/4 2/5 3/5 3/4 1/5 2/7 3/8 3/7

%e v \ . . . . . . . . . . .

%e 1/2 | 4/5 . . . . . . . . . .

%e 1/3 | 5/7 3/5 . . . . . . . . .

%e 2/3 | 7/8 9/11 12/13 . . . . . . . .

%e 1/4 | 2/3 7/13 11/14 8/17 . . . . . . .

%e 2/5 | 3/4 11/17 16/19 13/22 20/29 . . . . . .

%e 3/5 | 11/13 7/9 19/21 17/23 25/31 15/17 . . . . .

%e 3/4 | 10/11 13/15 17/18 16/19 23/26 27/29 24/25 . . . .

%e 1/5 | 7/11 1/2 13/17 3/7 5/9 5/7 19/23 5/13 . . .

%e 2/7 | 11/16 13/23 4/5 1/2 8/13 31/41 29/34 17/37 28/53 . .

%e 3/8 | 14/19 17/27 5/6 4/7 31/46 39/49 36/41 23/43 37/62 48/73 .

%e 3/7 | 13/17 2/3 23/27 19/31 29/41 9/11 33/37 11/19 7/11 9/13 21/29

%Y A348052 are the corresponding denominators.

%Y Cf. A007305, A007306, A348131, A348132.

%K nonn,frac,tabl

%O 2,1

%A _Hugo Pfoertner_, Sep 25 2021