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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.
7
1, 4, 7, 272, 211, 51012, 14197, 18640960, 1690981, 11225320100, 313968931, 10079828372880, 83828316391, 12627774819845668, 30436810578889, 21046391759976988928, 14425381885981321, 45032132922921758270916, 8649148282327007911, 120314227994702795221920400
OFFSET
1,2
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
KEYWORD
nonn,frac
AUTHOR
Amiram Eldar, Oct 01 2021
STATUS
approved