On the self matching properties of [jτ] Original Research Article

The graph of [jτ] against j displays self matching in that if we displace this graph by a distance of Fi, 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].