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

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..387

Wikipedia, Velocity-addition formula.

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A289381 A126577 A236288 * A073164 A297215 A134900

Adjacent sequences: A348128 A348129 A348130 * A348132 A348133 A348134

Keyword

nonn,frac

Author

Amiram Eldar, Oct 01 2021

Status

approved