  Integer compositions with certain colored parts were introduced by Andrews in
2007 to address a number-theoretic problem. Integer compositions allowing zero
as some parts were introduced by Ouvry and Polychronakos in 2019. We give a
bijection between these two varieties of compositions and determine various
combinatorial properties of these multicompositions. In particular, we
determine the count of multicompositions by number of all parts, number of
positive parts, and number of zeros. Then, working from three types of
compositions with restricted parts that are counted by the Fibonacci sequence,
we find the sequences counting multicompositions with analogous restrictions.
With these tools, we give combinatorial proofs of summation formulas for
generalizations of the Jacobsthal and Pell sequences.
