Exact solution of a minimal recurrence

In this note we nd the exact solution for the minimal recurrence S n = min bn=2c k=1 faS n?k + bS k g; where a and b are positive integers and a b. We prove that the solution is the same as that of the recurrence relation S n = aS dn=2e + bS bn=2c. In other words, S n = S 1 +(a+b?1)S 1 P n?1 i=1 a z(i) b blg nc?z(i) , where z(i) is the number of zeros in the binary representation of i. The proof follows from an interesting combinatorial property.