A study of a family of self-referential sequences

TL;DR

This paper introduces and analyzes a three-parameter family of self-referential integer sequences, revealing diverse behaviors including convergence, explicit formulas, periodicity, geometric patterns, and connections to meta-Fibonacci recurrences.

Contribution

It provides a comprehensive in-depth analysis of subfamilies of self-referential sequences, complementing existing studies and uncovering new properties and connections.

Findings

Sequences with y>z>0 converge to a positive root.

Explicit formulas for certain subfamilies as Beatty sequences.

Sequences with y=0 and z≥2 become periodic and satisfy linear recurrences.

Abstract

We introduce and analyze a three-parameter family of self-referential integer sequences $S(x,y,z)$: starting from $a(1)=x$, each term advances by $y$ when the index $k$ has already appeared as a value and by $z$ otherwise. This simple rule generates a surprising zoo of behaviors, many of which are catalogued - albeit in a rather unstructured fashion - in the OEIS. This family has recently and independently been studied by Fokkink and Joshi, who named them "hiccup sequences" and established their general morphic nature. Our work provides a complementary, in-depth analysis of major subfamilies. Whenever $y>z>0$, we prove that the density $a(k)/k$ converges to the positive root of $r^{2}-zr-(y-z)=0$. Two subfamilies, $S(x,Z+1,Z)$ and $S(x,Z,Z+1)$, yield explicit non-homogeneous Beatty sequences, providing explicit formulas for numerous OEIS entries. For $y=0$ and $z \ge 2$, the sequences eventually become periodic and satisfy linear recurrences. Critical cases with a zero discriminant unveil geometric patterns on triangular, square, and hexagonal lattices. Finally, via tree-like representations we uncover a tight link with meta-Fibonacci recurrences. These results position $S(x,y,z)$ as a unifying framework connecting additive combinatorics, number theory, and discrete dynamics.