This paper discusses complementary equations, such as a_n = b_n + b_2n, in which (a_n) and (b_n) are strictly increasing complementary sequences for which, in addition to the subscript n, at least one subscript in advance of n, such as n + 1 or 2n, occurs. Some of the equations are solved with proofs, and others are presented as conjectures or examples based on Mathematica programs that appear in the final section of the paper.