Counting arithmetic formulas

An arithmetic formula is an expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to as goes to infinity, solving a conjecture of E.K. Gnang and D. Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to and using exactly multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers , we compare the lengths of the arithmetic formulas for that each encoding produces with the length of the shortest formula for (which we estimate from below). We briefly discuss the time-space tradeoff offered by each.