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 n as n goes to infinity, solving a conjecture of E. K. Gnang and D. Zeilberger [5]. We give also an asymptotic formula for the number of arithmetic formulas evaluating to n and using exactly k multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers n, we compare the lengths of the arithmetic formulas for n that each encoding produces with the length of the shortest formula for n (which we estimate from below). We briefly discuss the time-space tradeoff offered by each.