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.

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

Offset: 1

Author

Amiram Eldar, Oct 01 2021

Keywords

• nonn
• frac

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

• Amiram Eldar, Table of n, a(n) for n = 1..387
• Wikipedia, Velocity-addition formula.

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