论文标题
Gerrymander序列或A348456
The gerrymander sequence, or A348456
论文作者
论文摘要
Kauers，Koutschan和Spahn最近宣布，所谓的{\ em Gerrymander sequence}的长度显着增加，在OEI中以A348456给出，将序列从3个术语扩展到7个术语。我们将进一步扩展到11个术语，但更明显地证明了该系数的增长为$λ^{4L^2}，$，其中$λ\约1.7445498，$，等于自我避免步行的相应数量，可以交叉一个正方形（WCAS），或者是自行的polygons crotsing a Square a Square（PCAS）。这些分别是OEIS序列A007764和A333323。因此，我们已经在这些先前独立的问题之间建立了密切的联系。 We have also related the sub-dominant behaviour to that of WCAS and PCAS, allowing us to conjecture that the coefficients of the gerrymander sequence A348456 grow as $λ^{4L^2+dL+e} \cdot L^g,$ where $d=-8.08708 \pm 0.0002,$ $e \approx 7.69$ and $g = 0.75 \pm 0.01，$ g $几乎肯定是$ 3/4 $。 我们还在相关的gerrymander多项式（定义）中产生了26个术语，并能够以令人满意的精度预测渐近行为。确实，它的行为与$ l $倍乘以广义Gerrymander序列的相应系数。 我们为计算这些序列提供的改进算法是我们最近开发的，用于扩展越过正方形或六边形晶格域的锯和SAPS的许多序列。它利用了最小的完美哈希功能和路径数量计数数组的原位内存更新。
Recently Kauers, Koutschan and Spahn announced a significant increase in the length of the so-called {\em gerrymander sequence}, given as A348456 in the OEIS, extending the sequence from 3 terms to 7 terms. We give a further extension to 11 terms, but more significantly prove that the coefficients grow as $λ^{4L^2},$ where $λ\approx 1.7445498, $ and is equal to the corresponding quantity for self-avoiding walks crossing a square (WCAS), or self-avoiding polygons crossing a square (PCAS). These are, respectively, OEIS sequences A007764 and A333323. Thus we have established a close connection between these previously separate problems. We have also related the sub-dominant behaviour to that of WCAS and PCAS, allowing us to conjecture that the coefficients of the gerrymander sequence A348456 grow as $λ^{4L^2+dL+e} \cdot L^g,$ where $d=-8.08708 \pm 0.0002,$ $e \approx 7.69$ and $g = 0.75 \pm 0.01,$ with $g$ almost certainly $3/4$ exactly. We also have generated 26 terms in the related gerrymander polynomial (defined below), and have been able to predict the asymptotic behaviour with a satisfying degree of precision. Indeed, it behaves exactly as $L$ times the corresponding coefficient of the generalised gerrymander sequence. The improved algorithm we give for counting these sequences is a variation of that which we recently developed for extending a number of sequences for SAWs and SAPs crossing a domain of the square or hexagonal lattices. It makes use of a minimal perfect hash function and in-place memory updating of the arrays for the counts of the number of paths.
