Consider the sequences of numbers $\left\{0, 1, 2\right\}$ with length $n$. There are $3^n$ such sequences. I define each sequence like a function. If a function consists of {0,1,2} elements of the length $«n»$, let's consider this function $\phi (n).$
Because, we can deduce all sequences for only finite number $n$, If we accept any infinity sequence equal to a specific function, problematic points occur. Because, we have infinitely number of function, which is we can not deduce all functions.
Definition: In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit.
For infinity sequence , which consist of elements $\left\{0,1,2\right\}$, if we have a specific $n-$th term any formula, we have a "specific mathematical function" (algebraic closed form expression or non-elementary function), which is we are looking for.
I can choose an infinitely number of functions that are Closed-Form Expression in term of a $"n".$
Examples: let $n\in\mathbb{Z^{+}} \bigcup \left\{0\right\}$
$$\phi(n)=n+2-3\lfloor \frac{n+2}{3}\rfloor=\left\{0,1,2,0,1,2\cdots \right\}$$
$$\phi(n)=n+1-3\lfloor \frac{n+1}{3}\rfloor=\left\{ 2,0,1,2,0,1\cdots \right\}$$
$$\phi(n)=n-3\lfloor \frac{n}{3}\rfloor=\left\{ 1,2,0,1,2,0\cdots\right\}$$
$$\phi(n)=n^n-3\lfloor \frac{n^n}{3}\rfloor=\left\{1,1,0,1,2,0,1,1,0,1,2,0\cdots \right\}$$
$$\phi(n)=n^n+n-3\lfloor \frac{n^n+n}{3}\rfloor=\left\{2,0,0,2,1,0,2,0,0,2,1,0\cdots \right\}$$
$$\phi(n)=\lfloor 10^n \pi \rfloor - 3 {\lfloor \frac{ \lfloor 10^n \pi \rfloor }{3}}\rfloor$$
$$\phi(n)=\lfloor 10^n e \rfloor - 3 {\lfloor \frac{ \lfloor 10^n e \rfloor }{3}}\rfloor$$
$$\phi(n)=\lfloor 10^n \sqrt2 \rfloor - 3 {\lfloor \frac{ \lfloor 10^n \sqrt2 \rfloor }{3}}\rfloor$$
$$\phi(n)=\lfloor 10^n \log \pi \rfloor - 3 {\lfloor \frac{ \lfloor 10^n \log \pi \rfloor }{3}}\rfloor$$
$$\cdots \cdots \cdots \cdots \cdots$$
For these periodic and non-periodic sequences there are exist $n-$th term "closed-form expression."
Then, we can define an infinitely number of "specific mathematical functions", (non-elementary, non-algebraic) which is non-periodic.
Example:
$$ \phi(n)=\lfloor 10^n \displaystyle\int_0^\infty e^{-x^n}dx \rfloor - 3 {\lfloor \frac{ \lfloor 10^n \displaystyle\int_0^\infty e^{-x^n}dx \rfloor }{3}}\rfloor$$
$$\cdots \cdots \cdots \cdots \cdots$$
Claims: (only for infinitely number sequences)
A) There exist infinitely number of sequences that, consist of elements $\left\{0,1,2\right\}$, which is can not express by the any "closed-form expression" or any "specific mathematical function".
B) There exist infinitely number of sequences that, consist of elements $\left\{0,1,2\right\}$, which is can express by the any closed-form expression.
C) There exist infinitely number of sequences that, consist of elements $\left\{0,1,2\right\}$, which is can express by the any "specific mathematical function" (non-elementary, non-algebraic).
Which of these claims are true? I'm looking for a proof that confirms or denies the claims.
Thank you!