We display sequences of interest that do or do not satisfy a Nicomachean identity, and create from them new sequences that do. This is done by introducing translation parameters, adjoining variable parameters, and forming sequential ("Mason-Liouvillian") products. Degeneracies in this ostensibly cubic framework enable us to employ facts about both definite and indefinite quadratic forms. We prove new results about gaps in Nicomachean sequences, and display novel Nicomachean sequences formed from Fibonacci numbers. We also briefly illustrate connections with certain elliptic curves and their geometry. This paper can be viewed as an in-depth study of variations of sequence A000537.