 
Summary: SOLUTION TO PROBLEM 10663
PROPOSED BY E. DEUTCH
Shalosh B. Ekhad and Tewodros Amdeberhan
Temple University, Mathematics and
DeVry Institute, Mathematics
630 US Highway One, North Brunswick, NJ 08902
amdberhan@admin.nj.devry.edu
PROBLEM: P The Pell sequence is de ned by p0 = 1;p1 = 2; and pn = 2pn,1 + pn,2 for
n 2. Show that, for n 0,
pn =
X i + j + k!
i!j!k! ;
where the summation extends over all nonnegative integers i;j;k satisfying i+ j + 2k = n.
PROOF: Replacing i+ j + k = n ,k, i = n,2k ,j and denoting the sum in above by an, we
have
1 an =
X
k
n,k
