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

Kimberling, Clark

Fibonacci Q. 33, No. 1, 3-8 (1995).

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.

Reviewer: J.Piehler (Merseburg)

MSC:

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

Keywords:

Fibonacci numbers; representation; Zeckendorf array; Wythoff array

Citations:

Zbl 0531.05018