Representations of U(3) in U(N). Nature of problem: U(N) -> U(3) plethysm, that is, finding the complete set of irreducible representations (irreps) of U(3) in specific irreps of U(N) where N=(n+1) (n+2)/2 for nonnegative integer n values. Solution method: Solutions are obtained by applying a simple difference algorithm to the U(3) weight distribution function. The latter is generated in three steps: 1) by indentifying the N levels of U(N) as the distinguishable arrangements of n oscillator quanta in three cartesian directions, 2) by distributing the total number of qaunta (n * m if m is the number of valence particles) among these levels subject to restrictions (betweeness conditions) of the Gelfand scheme for labeling basis states of U(N), and 3) by summing over all the N levels to determine the final distribution of quanta in the three cartesian directions.