Summary: Increasing sequences $a$( ) and $b$( ) that partition the sequence of positive integers are called complementary sequences, and equations that explicitly involve both $a$( ) and $b$( ) are called complementary equations. This article surveys several families of such equations, including $b(n) = a(jn) +- r, b(n) = a(jn)+kn, b(n) = f(a(n))$, and $b(n) = a(b(n-1))+qn+r$.

11B37

Beatty sequence, complementary equation, complementary sequences, dispersion, inverse, polygonal numbers, stolarsky array, wythoff array, wythoff difference array, wythoff sequences