Numerical Analogues of Aronson's Sequence

TL;DR

This paper explores numerical analogues of Aronson's sequence, defining new sequences based on parity and positional conditions, and investigates their properties and generalizations, some of which are novel contributions.

Contribution

It introduces new numerical sequences inspired by Aronson's sequence, analyzing their properties and providing generalizations that deepen understanding of related integer sequences.

Findings

The sequence a(n) is the smallest positive integer greater than a(n-1) with specific parity conditions.

The sequence a^(2)(n) = a(a(n)) equals 2n+3 for n >= 1.

Several new generalizations of Aronson's sequence are proposed and analyzed.

Abstract

Aronson's sequence 1, 4, 11, 16, ... is defined by the English sentence ``t is the first, fourth, eleventh, sixteenth, ... letter of this sentence.'' This paper introduces some numerical analogues, such as: a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition ``n is a member of the sequence if and only if a(n) is odd.'' This sequence can also be characterized by its ``square'', the sequence a^(2)(n) = a(a(n)), which equals 2n+3 for n >= 1. There are many generalizations of this sequence, some of which are new, while others throw new light on previously known sequences.