Efficient Calculation the Number of Partitions of the Set $\{1, 2, \ldots, 3n\}$ into Subsets $\{x, y, z\}$ Satisfying $x+y=z$

TL;DR

This paper develops criteria for efficiently counting partitions of the set {1, 2, ..., 3n} into triplets where two elements sum to the third, and enumerates such partitions for specific values, expanding known integer sequences.

Contribution

It introduces new criteria for pruning the search space and enumerates all such partitions for n=16 and n=17, contributing new terms to OEIS sequence A108235.

Findings

Enumerated partitions for n=16 and n=17.

Added new terms to OEIS sequence A108235.

Developed criteria for efficient partition enumeration.

Abstract

Consider the set $\{1,2,\ldots,3n\}$. We are interested in the number of partitions of this set into subsets of three elements each, where the sum of two of them equals the third. We give some criteria such a partition has to fulfill, which can be used for efficient pruning in the search for these partitions. In particular, we enumerate all such partitions for $n=16$ and $n=17$ adding new terms to the series A108235 in the Online Encyclopedia of Integer Sequences.