  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 a^k(n)=a(a(...(a(n))...)) goes
to 1 in finitely many steps and we call the pattern
(n,a(n),a^2(n),...,a^k(n)=1) a jumping pattern from n down to 1. In this paper
we look at jumping sequences which 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)/i^2 then the cost minimizing
jump sequence has the property that the number m satisfies m=a^q(p) for
arbitrary q and some p (depending on q) if and only if m is a Pell number.
