# Is $a_n = f(a + b c^n)$ when $a_0 = \ln(2)$ and $a_{n+1} = (e+1) a_n (a_n - 1)$?

Definition

Let $a_n$ be a real sequence.

Assume there exists a continuous real-periodic function $f(x)$ such that

$f(n) = a_n$

And $f(x)$ has the period $t$ , where $t$ is An irrational real number.

Then we say $a_n$ is periodic and has An irrational period. Or we say $a_n$ Has An irrational periodic orbit.

Let $a_0 = \ln(2)$ and $a_{n+1} = (e+1) a_n (1 - a_n)$.

Does there exist an $f$ such that :

$$a_n = f( a + b c^n)$$

Where $a,b,c$ are (fixed) real values and $f$ is a real-continuous periodic function with real irrational period $t$ ?

• I changed " is " $a_n = f$ ... To : Does there exist An $f$ such that $a_n = f$ ... Although imho the " is " was sufficient to be clear and consistent. Not sure if that improves , but it can not be worse I assume. Thanks to hippocampus from chat for the suggestion.
– mick
Dec 13, 2015 at 21:49
• That is hippalectryon. Not hippocampus. en.m.wikipedia.org/wiki/Seahorse
– mick
Dec 13, 2015 at 22:05