Counting with Hiccup Sequences

Source 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 $x\geq3$ the gaps between its missing values are $1,2,3,\ldots$, 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 $z\geq2$ and every initial value $x\geq2$. Further interpretations involve polyomino vertices, monochromatic pairs, and, by a theorem of Bickle, Ramsey core numbers.