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COMMENT
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An effort to model the function of hyperbolic-type numbers :
f[n_] := If[n == 0, 0, Sinh[n*Pi]/Sinh[Pi]];
out2 = Table[Floor[f[n]], {n, 0, 20}]
{0, 1, 23, 536, 12414, 287287, 6648038, 153840223, 3559969319, 82380155808,
1906333864611, 44113886016440, 1020825877143916, 23622617854574666,
546643738954316017, 12649714742775054564, 292723160754894011024,
6773816689524637485291, 156750809963080035635001, 3627322313294829420472342,
83938750731977514666561352}
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