m-Bonacci graceful labeling

Diffractive m-bonacci lenses.

Fibonacci zone plates are proving to be promising candidates in image forming devices. In this letter we show that the set of Fibonacci zone plates are a particular member of a new family of diffractive lenses which can be designed on the basis of a given m-bonacci sequence. These lenses produce twin axial foci whose separation depends on the m-golden mean. Therefore, with this generalization, ...

Balances of m-bonacci Words

The m-bonacci word is a generalization of the Fibonacci word to the m-letter alphabet A = {0, . . . ,m − 1}. It is the unique fixed point of the Pisot–type substitution φm : 0 → 01, 1 → 02, . . . , (m − 2) → 0(m − 1), and (m − 1) → 0. A result of Adamczewski implies the existence of constants c such that the m-bonacci word is c-balanced, i.e., numbers of letter a occurring in two factors of the...

Graceful Tree Labeling

Graceful and odd graceful labeling of some graphs

In this paper, we prove that the square graph of bistar Bn,n, the splitting graph of Bn,n and the splitting graph of star K1,n are graceful graphs. We also prove that the splitting graph and the shadow graph of bistar Bn,n admit odd graceful labeling.

Ambiguity in the m-Bonacci Numeration System

akGk, where ak ∈ N0, see for instance [8]. Some sequences (Gk)k∈N have even nicer property: Every positive integer can be expressed as a sum of distinct elements of the sequence (Gk)k∈N. The necessary and sufficient condition so that it is possible is that the sequence satisfies G1 = 1 and Gn −1 ≤ ∑n−1 i=1 Gi for all n ∈ N. Example of such a sequence is (2)k≥1 or (Fk)k≥1, the sequence of Fibona...