Fibonacci words in hyperbolic Pascal triangles

Abstract

The hyperbolic Pascal triangle ${\mathcal{H}\mathcal{P}\mathcal{T}}_{4,q}$ $(q\ge 5)$ is a new mathematical construction, which is a geometrical generalization of Pascal's arithmetical triangle. In the present study we show that a natural pattern of rows of ${\mathcal{H}\mathcal{P}\mathcal{T}}_{4,5}$ is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the hyperbolic Pascal triangles. The geometrical properties of a ${\mathcal{H}\mathcal{P}\mathcal{T}}_{4,q}$ imply a graph structure between the finite Fibonacci words.

10 pages, 4 figures, Acta Univ. Sapientiae, Mathematica, 2017