# arithmetic geometry – How to find a continuous or discontinuous function \$f\$ which passes through points and satisfies other conditions?

Hello everyone and nothing,

How to find a continuous function $$f$$ for a real $$x :f (x)$$ or discontinuous for an integer $$n :f (n)$$.

Such as :

$$f (0) = 0, f (1) = 1, f (2) = 36, f (3) = 1179, f (4) = 38346, f (5) = 1246285, f (6) = 40504909, f (7) = 1316424317 f (8) = 42784149984.$$

And $$f$$ is increasing then decreasing.

And

$$lim_{x rightarrow + infty} f(x) = 0$$

or

$$lim_{n rightarrow + infty} f(n) = 0$$

$$f (x) = ??$$

If $$f (n)$$ is a sequence of zero limit increasing from $$0$$ then decreasing towards $$0$$ what would be the sequence $$f (9) =?, f (10) =?, f (11) =?, …f (n) = ??.$$