Hyperbolic Fibonacci Sequence

Fibonacci words, hyperbolic tilings and grossone

In this paper, we study the contribution of the theory of grossone to the study of infinite Fibonacci words, combining this tool with the help of a particular tiling of the hyperbolic plane : the tiling {7, 3}, called the heptagrid. With the help of the numeral system based on grossone, we obtain a richer family of infinite Fibonacci words compared with the traditional approach. ACM-class: F.2....

Fibonacci words in hyperbolic Pascal triangles

The hyperbolic Pascal triangle HPT 4,q (q ≥ 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 HPT 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 h...

The Fibonacci Sequence Mod m

We know that ( ) mod n F p forms a periodic sequence (vide Theorem 4). Let ( ) h p denote the length of the sequence. Let p be a prime such that: { } ( ) 2,3 mod 5 p ≡ a sufficient and necessary condition to ensure that ( ) 2 2 h p p + . We shall denote this group 1 G F . Let { } 1, 2 , , k D d d d = be the non-empty set of k divisors of 2 2 p + . Then for ( ) [ ] 1 min G i F h p d = such that ...

Puzzle Geometry and Rigidity: the Fibonacci Cycle Is Hyperbolic

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and “complex bounds”, two generalized polynomial-like maps which admits a topological conjugacy, quasiconformal outside the filled-in Julia set are, indeed, quasiconformally conjugated. The proof uses a new abstract remova...

Periods and Entry Points in Fibonacci Sequence