Number of instances of SAT (boolean satisfiability) problems of size N?
If you consider 3-Conjunctive Normal Form (3CNF) (every clause has exactly three literals) and you allow clauses with repeated variables, then it is easy to see that using $N$ variables the number of different 3SAT clauses is $(2N)^3$
($2N$ because every literal can appear unnegated or negated).

So the total number of distinct 3CNF formulas (without repeated clauses) is $2^{(2N)^3}$.

But be careful because the "size" of a 3CNF formula (e.g. considered as an input of a decision problem) is usually the size of its representation. For example, using alphabet $\Sigma = \{0,...,9,``,",-\}$
the 3CNF formula $(x_1\lor\bar{x}_2\lor x_3)\land (\lor x_2 \lor \bar{x}_3 \lor \bar{x}_4)$ can be represented with the string:

1,-2,3,2,-3,-4


and its length is 14.