  In a recent article the first three authors proved that in dimension $4m+1$
all homotopy spheres that bound parallelizable manifolds admit Einstein metrics
of positive scalar curvature which, in fact, are Sasakian-Einstein. They also
conjectured that all such homotopy spheres in dimension $4m-1, m\geq2$ admit
Sasakian-Einstein metrics \cite{BGK}, and proved this for the simplest case,
namely dimension $7.$ In this paper we describe computer programs that show
that this conjecture is also true for 11-spheres and 15-spheres. Moreover, a
program is given that determines the partition of the 8610 deformation classes
of Sasakian-Einstein metrics into the 28 distinct oriented diffomorphism types
in dimension $7.$
