Jumping Sequences

An integer sequence a(n) is called a jump sequence if a(1) = 1 and 1 ≤ a(n) < n for n ≥ 2. Such a sequence has the property that ak(n) = a(a(· · · (a(n)) · · · )) goes to 1 in finitely many steps. We call the pattern (n, a(n), a2(n), . . . , al(n) = 1) a jumping pattern from n down to 1. In this paper we look at jumping sequences that are weight minimizing with respect to various weight functions (where a weight w(i, j) is given to each jump from j down to i). Our main result is to show that if w(i, j) = (i + j)/i2, then the cost-minimizing jump sequence has the property that the number m satisfies m = aq(p) for arbitrary q and some p (depending on q) if and only if m is a Pell number.