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The Zeckendorf array equals the Wythoff array. (English) Zbl 0828.11009

It is known that every natural \(n\) is uniquely a sum of nonconsecutive Fibonacci numbers, the Zeckendorf representation of \(n\). This representation can be arranged in the Zeckendorf array \(Z(i,j)\) where the column \(j\) is the increasing sequence of all \(n\) the least term in its Zeckendorf representation being \(F_{j + 1}\). The Wythoff array \(W(i,j)\) was introduced by D. R. Morrison [The Fibonacci sequence, Collect. Manuscr., 18th Anniv. Vol., 134-136 (1980; Zbl 0531.05018)]. The author now shows that the Zeckendorf array equals the Wythoff array and gives some other results concerning generalized Zeckendorf arrays.

MSC:

11B37 Recurrences
05A10 Factorials, binomial coefficients, combinatorial functions

Citations:

Zbl 0531.05018
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