Algorithms for complementary sequences

TL;DR

This paper develops methods to compute the $n$-th positive integer that does not satisfy certain mathematical conditions, providing explicit formulas for various complementary sequences such as non-$k$-gonal and non-$k$-th powers.

Contribution

It introduces novel formulas and approaches, including fixed point and bisection methods, for calculating complementary sequences with complex constraints.

Findings

Derived explicit formulas for non-$k$-gonal numbers.

Provided formulas for non-$k$-gonal-pyramidal and non-$k$-simplex numbers.

Extended methods to compute non-$k$-th powers and related sequences.

Abstract

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, and non-$k$-th-powers. For example, we show that the $n$-th non $k$-gonal number is given by $n+\text{round}\left(\sqrt{\frac{2n-2+\left\lfloor\frac{k+1}{4}\right\rfloor}{k-2}}\right)$ and that the $n$-th non-second-hexagonal number is $n+\left\lceil\sqrt{\frac{n}{2}}\right\rceil-1$.