Do all increasing sequences that only consist of natural numbers have at least one general formula? Do all increasing sequences that only consist of natural numbers have at least one general formula? I am wondering if this is true or not.
If floor function and ceiling functions are allowed to be in the general formula, then because infinitely many lines could be drawn in the first quadrant of a Descartes plane, infinitely many sequences could be represented then. 
 A: First you have to define what counts as a "general formula".
But unless you have a very unusual idea of what a formula is, anything that has a "general formula" will be computable by a Turing machine.
There are only countably many Turing machines. But there are uncountably many increasing sequences of natural numbers.
A: It's not true at all!
Take the sequence of prime numbers for example.
A: Increasing sequence of natural numbers can be polynomials, expenentials $2*n, 3^n$ or linear combination of these.
(1)-If the mth-difference of the sequence is zero then the nth term is a polynomial of degree $(m-1)$ $T_n=f_m(n)$. E.g., Third difference is zero then $T_n=An^2+Bn+C$
(2)-If mth difference become a G.P of common ration $r$, then the nth term is a polynomial of degree $(m-1)$ plus and exponential like $kr^n$. The second difference becomes a GP, then
$T_n=An^2+BN+C+Dr^n$.
(3)-If the swqience follows: $A T_n + BT_{n-1}+C T_{n-2}=0$, Then $T_n=C_1 a^n +C_2 b^n$, where $a,b$ are the roots os $Ax^2+Bx+C=0$.
Note: Somehow examiners follow an unwritten rule that the nth term should be as one of the three case as above.
A: No. For example, I define the following sequence. The first element is drawn randomly from the numbers 1 to 10. The second number is the first number plus a number drawn randomly from the numbers 1 to 10. The third number is the second number plus a number drawn randomly from the numbers 1 to 10 and so on and so forth.
If the numbers are really random, there is no forumula to describe the sequence. If the number generator is pseudo-random, then of course, there will be a formula.
