Counting with Hiccup Sequences
Abstract
The $(0,x,1,2)$-hiccup sequence starts at x and increases by 1 when the current index has already appeared as a value, and by 2 otherwise. We prove directly that for x≥3 the gaps between its missing values are 1,2,3,…, each repeated x−2 times, and we derive closed formulas, an explicit error bound, and a generating function in terms of Ramanujan's theta functions. The counting functions of the complements satisfy nested recurrences. The sequences of missing values are the partial complements of polygonal numbers. In particular, the (x−2)-fold composition of the enumeration of the positive integers that are not x-gonal is the hiccup sequence minus 1. We extend the nested recurrence and a tree interpretation to every miss increment z≥2 and every initial value x≥2. Further interpretations involve polyomino vertices, monochromatic pairs, and, by a theorem of Bickle, Ramsey core numbers.
Related
BibTeX
@misc{cloitre2026counting,
title = {{Counting with Hiccup Sequences}},
author = {Benoit Cloitre},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13908},
eprint = {2610.13908},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13908},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counting with Hiccup Sequences
AU - Benoit Cloitre
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13908
UR - https://arxiv.org/abs/2610.13908
AB - The $(0,x,1,2)$-hiccup sequence starts at x and increases by 1 when the current index has already appeared as a value, and by 2 otherwise. We prove directly that for x≥3 the gaps between its missing values are 1,2,3,…, each repeated x−2 times, and we derive closed formulas, an explicit error bound, and a generating function in terms of Ramanujan's theta functions. The counting functions of the complements satisfy nested recurrences. The sequences of missing values are the partial complements of polygonal numbers. In particular, the (x−2)-fold composition of the enumeration of the positive integers that are not x-gonal is the hiccup sequence minus 1. We extend the nested recurrence and a tree interpretation to every miss increment z≥2 and every initial value x≥2. Further interpretations involve polyomino vertices, monochromatic pairs, and, by a theorem of Bickle, Ramsey core numbers.
ER -