  We construct, for each 2<r<18, an explicit family of higher Chow cycles of
type (2,1) on a family of lattice-polarized K3 surfaces of generic Picard rank
r, and prove that the indecomposable part of this cycle is non-torsion for very
general members of the family. These are the first explicit examples of such
families in middle Picard rank. Our construction is based on singular double
plane model of K3 surfaces, and the proof of indecomposability is done by a
degeneration method.
