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COMMENTS
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The hyperbolic numbers are bigger:
f[n_] := If[n == 0, 0, Sinh[n*Pi/4]/Sinh[Pi/4]];
out2 = Table[Floor[f[n]], {n, 0, 20}]
{0, 1, 2, 6, 13, 29, 64, 140, 308, 676, 1482, 3251, 7132, 15643, 34310,
75253, 165051, 362004, 793977, 1741415, 3819411}
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