Tchoukaillon数组的渐近性质与Beluhov猜想
Asymptotics of the Tchoukaillon array and a conjecture of Beluhov
AI总结：
本文证明了Beluhov关于Tchoukaillon数组一般项渐近性质的猜想，建立了更强的一致估计，并给出了水平区域的结构结论与定位整数的O(√M)算法。
AI中文摘要：
Tchoukaillon数组是由一行Mancala单人棋游戏产生的正整数无限数组，每个正整数恰好出现一次。其第0列是Flavius Josephus筛，第0行是Tchoukaillon数序列；这两条边的渐近性质是Andersson以及Broline和Loeb的经典结果。基于数值证据，N. Beluhov（由Knuth转述）提出猜想：当i,j→∞时，一般项T_{i,j}满足T_{i,j}≈(πi+2j)²/(4π)。我们证明了该猜想，实际上建立了更强的一致估计：T_{i,j}=(πi+2j+2)²/(4π)+O((i+j+1)^{4/3})，其中常数π和2由数组自身通过Wallis乘积的递归关系产生，独立于两条边定理。等价地，该项的平方根渐近线性：√T_{i,j}=(√π/2)i+(1/√π)(j+1)+O((i+j+1)^{1/3})，是两条边增长率的线性组合。作为推论，我们得到水平区域{T_{i,j}≤V}是三角形，仅存在宽度为O(V^{1/6})的边界，以及一个O(√M)的算法，可定位给定整数M所在的行和列。
英文摘要：
The Tchoukaillon array is an infinite array of the positive integers, arising from a one-row Mancala solitaire, in which each positive integer occurs exactly once. Its zeroth column is the Flavius Josephus sieve and its zeroth row is the sequence of Tchoukaillon numbers; the asymptotics of these two edges are classical results of Andersson and of Broline and Loeb. On the basis of numerical evidence, N. Beluhov conjectured (as relayed by Knuth) that the general entry $T_{i,j}$ satisfies $T_{i,j} \approx (πi+2j)^2/(4π)$ as $i,j \to \infty$. We prove this conjecture. In fact we establish the stronger uniform estimate $T_{i,j} = (πi+2j+2)^2/(4π) + O((i+j+1)^{4/3})$, in which both constants $π$ and $2$ are produced by the array's own recursion through a Wallis product, independently of the two edge theorems. Equivalently, the square root of the entry is asymptotically linear, $\sqrt{T_{i,j}} = (\sqrtπ/2)\, i + (1/\sqrtπ)(j+1) + O((i+j+1)^{1/3})$, the linear blend of the two edge growth-rates. As corollaries we obtain that the level regions $\{T_{i,j} \le V\}$ are triangles up to a boundary of width $O(V^{1/6})$, and an $O(\sqrt{M})$ algorithm that locates the row and column of a given integer $M$.
