Algorithms for Complementary Sequences

Finding the n-th positive square number is easy, as it is simply $n^2$. But how do we find the complementary sequence, i.e., the n-th positive non-square number? For this case there is an explicit formula. However, for general constraints on numbers, a formula is harder to find. In this paper, we study how to compute the n-th integer that does (or does not) satisfy a certain condition. In particular, we consider it as a fixed point problem, relate it to the iterative method of Lambek and Moser, study a bisection approach to this problem, and provide novel formulas for various complementary sequences including the non-k-gonal numbers, non-k-gonal-pyramidal numbers, non-k-simplex numbers, non-sum-of-k-th-powers, non-k-th-powers, and non-Jacobsthal numbers.