AI 中文总结
研究顶点数比边数多一的(3,1)-正则图，通过将其计数序列表示为三重和对角线、留数表示、图族组合递推这三种方式，确认Kauers和Koutschan猜测的递推关系，三种方式均涉及计算机计算并给出形式完整证明。
AI 中文摘要
OEIS的序列A339987通过顶点数的一半来计数顶点数比边数多一的(3,1)-正则图。Kauers和Koutschan在2023年猜测了该序列满足的递推关系。我们通过三种方式进行确认：一是表示为三重和的对角线以及传统创造性消去法的精细变体以便进行事后验证；二是通过留数表示和基于约简的创造性消去法直接计算；三是通过图族上的组合递推和微分消去法计算。这三种方法都能得到形式上完整的证明且都以某种方式涉及计算机计算。
英文摘要
Sequence A339987 of the OEIS counts (3,1)-regular graphs having one more vertex than edges by half the number of vertices. A recurrence relation satisfied by this sequence was guessed by Kauers and Koutschan in 2023. We confirm it in three ways: first, by a representation as the diagonal of a triple sum and an elaborate variant of traditional creative telescoping that makes an a posteriori validation possible; second, by a residue representation and a direct calculation by reduction-based creative telescoping; third, by a combinatorial recurrence on graph families and a calculation by differential elimination. Each of those three approaches leads to a formally complete proof and involves a computer calculation in one way or another.
