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 $f (9) =?, f (10) =?, f (11) =?, …f (n) = ??.$