On the self matching properties of [jτ]

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• 作者：Bunder ; Martin ; Tognetti ; Keith
• 关键词：Fibonacci ; Zeckendorf ; Self matching ; Bernoulli integer sequences
• 刊名：Discrete Mathematics
• 出版年：2001
• 出版时间：October 28, 2001
• 年：2001
• 卷：241
• 期：1-3
• 页码：139-151
• 全文大小：106 K

文摘

The graph of [jτ] against j displays self matching in that if we displace this graph by a distance of F_i, then it is found that the displaced graph matches the original graph except at certain isolated points represented by an interesting Fibonacci function. From this it is shown that the frequency of mismatches is the unexpectedly simple expression 1/(τⁱ). The results are proved using lemmas, based on Zeckendorf sums, which have an appeal of their own. These also give simplified solutions to the recurrence of Downey and Griswold. Similar results apply with the Golden Sequence whose jth term is [(j+1)τ]−[jτ].