Tootfinder

My first submission to OEIS, although mostly the work by Markus Krötzsch.
https://oeis.org/A387334

On Cloitre's hiccup sequences
Robbert Fokkink, Gandhar Joshi
https://arxiv.org/abs/2507.16956 https://arxiv.org/pdf/2507.16956

On Cloitre's hiccup sequences
In 2003, Benoit Cloitre entered a bunch of sequences in the OEIS that we call \emph{hiccup} sequences. We collect the various claims, observations, and proofs of properties of these sequences that have been entered in the OEIS over the years, and present a unified approach, guided by a remarkable theorem of Bosma, Dekking, and Steiner.

Bounds for sets of remainders
Omkar Baraskar, Ingrid Vukusic
https://arxiv.org/abs/2508.20853 https://arxiv.org/pdf/2508.20853

Bounds for sets of remainders
Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has barely been studied. First, we prove that $s(n) = c \cdot n + O(n/(\log n \log \log n))$, where $c$ is an explicit constant. Then we focus on differences between consecutive terms $s(n)$ and $s(n+1)$. It turns out that the value can always increase by at most o…

A study of a family of self-referential sequences
Benoit Cloitre
https://arxiv.org/abs/2506.18103 https://arxiv.org/pdf/2506.18103

A study of a family of self-referential sequences
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. Whenever $y>z>0$, we prove that the density $a(k)/k$ converges to the positive root of $r^2 - z r - (y - z) = 0$. Two subfamilies,…

Non-orientable Nurikabe
Joseph Breen, Emma Copeland
https://arxiv.org/abs/2506.12612 https://arxiv.org/pdf/2506.12612

Non-orientable Nurikabe
We study Nurikabe puzzles on non-orientable surfaces. Specifically, we propose two versions of non-orientable Nurikabe and investigate their combinatorics on Möbius strips, Klein bottles, and projective planes of size $1\times n$. Our results establish new connections among the OEIS sequences A101946, A213387, A123203, and A001045 (the Jacobsthal sequence).