论文标题
$ s $ -E-LEGAL索引差异序列的复发关系
Recurrence Relations for $S$-Legal Index Difference Sequences
论文作者
论文摘要
Zeckendorf的定理意味着斐波那契数$ f_n $是最小的正整数，无法将其写入以前的非连续性斐波那契数字的总和。 Catral等。研究了斐波那契序列的变化，斐波那契被子序列：使用斐波那契螺旋瓷砖，并将整数分配到螺旋形的正方形中，使每个正方形包含最小的正无integer，无法表示为非试名的先前条款的总和。在每个正方形的索引的差异中，本质上捕获了这种邻接：$ i $ th和$ j $ -th的正方形在且仅当$ | i-j |时才相邻。 \ in \ {1，3，4 \} $或$ \ {i，j \} = \ {1，3 \} $。 我们考虑了这种结构的概括：鉴于一组积极的整数$ s $，$ s $ legal索引差异（$ s $ lid）序列$ $（a_n）_ {n = 1}^\ infty $来定义$ a_n $，以$ a_n $为$ \ ell y y y y y y y \ el f \ el f \ y \ y \ y \ y \ y l \ y; \ subset [n -1] $ with $ | i -j | \ notin s $ for l $ i，j \ in L $。我们讨论了$ S $ lid序列增长的结果，并证明了许多套装$ s $ s $ s $ s $ s $ lid序列的序列，这些序列遵循简单的复发关系。
Zeckendorf's Theorem implies that the Fibonacci number $F_n$ is the smallest positive integer that cannot be written as a sum of non-consecutive previous Fibonacci numbers. Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the $i$-th and $j$-th squares are adjacent if and only if $|i - j| \in \{1, 3, 4\}$ or $\{i, j\} = \{1, 3\}$. We consider a generalization of this construction: given a set of positive integers $S$, the $S$-legal index difference ($S$-LID) sequence $(a_n)_{n=1}^\infty$ is defined by letting $a_n$ to be the smallest positive integer that cannot be written as $\sum_{\ell \in L} a_\ell$ for some set $L \subset [n-1]$ with $|i - j| \notin S$ for all $i, j \in L$. We discuss our results governing the growth of $S$-LID sequences, as well as results proving that many families of sets $S$ yield $S$-LID sequences which follow simple recurrence relations.
