Trimer Covers in the Triangular Grid — Problem 11

Definitions and mathematical statement

For an integer $n\geq 1$, the supplied source writes \[ T(n):=T_{112}(3n-1,3n). \] The requested problem is to find a formula for $T(n)$, valid for every integer $n\geq 1$. The record does not define $T_{112}(a,b)$: in particular, it does not specify the underlying triangular-grid region, the permitted trimers, or the configurations being counted. Thus this notation identifies the source's target but does not give a self-contained definition of it.

The source additionally reports the apparent property that $T(n)$ has no prime factor greater than or equal to $4n$. Under the intended integer-valued interpretation of $T(n)$, this assertion is \[ \forall n\in{\mathbb{Z}}_{\geq 1}\;\forall p\in{\mathbb{Z}}_{\geq 2},\qquad \left(p\text{ is prime}\ \land\ p\mid T(n)\right)\Longrightarrow p<4n. \] This is a supplied empirical observation, not an established conclusion. The explicit request is for a formula; the record does not clarify whether proving this prime-factor assertion is a separate requested task. A complete mathematical formulation requires the missing definition of $T_{112}$.

Short English statement

Find a formula, valid for every integer $n\geq 1$, for the source quantity $T_{112}(3n-1,3n)$. The source observes that its prime factors appear to be smaller than $4n$. The underlying counting function is undefined in the supplied record, so this draft requires clarification and human mathematical review.

Sources

• Local problem registry, permanent UnsolvedMath ID $800352$, exact source problem code catalogue entry; supplied complete record titled Trimer Covers in the Triangular Grid --- Problem 11. The registry attributes the problem to Propp (2022) and labels it open; that status is supplied metadata, not independently verified here.
• Propp, Trimer Covers in the Triangular Grid: Twenty Mostly Open Problems (2022), source item recorded as "Problem 11 (#4)". Supplied source URL: https://arxiv.org/abs/2206.06472. The provenance records access on 2026-10-16 and extraction method arxiv_latex; these are supplied provenance details, not evidence of independent consultation for this draft.
• OEIS entry A364483, explicitly referenced in the supplied statement: https://oeis.org/A364483. The entry's contents and any formula it may contain were not supplied or independently verified here.