On the number of arithmetic formulas

For each positive integers n, let g(n) be the number of arithmetic expressions evaluating to n and involving only the constant 1, additions and multiplications, with the restriction that multiplication by 1 is not allowed. We consider two arithmetic expressions to be equal if one can be obtained from the other through a repeated application of the commutative and associative properties. We give an algorithm to compute g(n) and prove that log(g(n)) = βn + O(√n), as n → +∞, where β := log(24)/24.