We allow leading 0’s so the representation is not unique. More generally, we do not impose any additional constraints on the representations of numbers. Thus, while the standard Fibonacci representation of 11 = F2 + F3 + F4 + F5 is 10100 we will also admit 1111. A set X of natural numbers is recognizable in base U or U-recognizable if there is a finite state machine that accepts those words ove...

In the language of mathematical chemistry, Fibonacci cubes can be defined as the resonance graphs of fibonacenes. Lucas cubes form a symmetrization of Fibonacci cubes and appear as resonance graphs of cyclic polyphenantrenes. In this paper it is proved that the Wiener index of Fibonacci cubes can be written as the sum of products of four Fibonacci numbers which in turn yields a closed formula f...

The Fibonacci dimension fdim(G) of a graph G was introduced in [7] as the smallest integer d such that G admits an isometric embedding into Qd, the d-dimensional Fibonacci cube. A somewhat new combinatorial characterization of the Fibonacci dimension is given, which enables more comfortable proofs of some previously known results. In the second part of the paper the Fibonacci dimension of the r...

We introduce a matricial Toeplitz transform and prove that the Toeplitz transform of a second order recurrence sequence is another second order recurrence sequence. We investigate the injectivity of this transform and show how this distinguishes the Fibonacci sequence among other recurrence sequences. We then obtain new Fibonacci identities as an application of our transform.

Functional programmers happily use equational reasoning and induction to prove properties of recursive programs. To show properties of corecursive programs they employ coinduction, per perhaps less enthusiastically. Coinduction is often considered as a rather low-level proof method, especially, as it seems to depart rather radically from equational reasoning. In this talk we introduce an altern...

In this paper, we define five parameters generalization of Fibonacci numbers that generalizes Fibonacci, Pell, Modified Jacobsthal, Narayana, Padovan, k-Fibonacci, k-Pell, k-Jacobsthal and p-numbers, distance numbers, (2, k)-distance generalized (k, r)-Fibonacci in the sense by extending definition a recurrence relation with two adding three distance, simultaneously. Tiling combinatorial interp...

Each solution {xn} of the equation in the title is either eventually periodic with period 3 or else, it converges to zero—which case occurs depends on whether the ratio of the initial values of {xn} is rational or irrational. Further, the sequence of ratios {xn/xn−1} satisfies a first-order difference equation that has periodic orbits of all integer periods except 3. p-cycles for each p = 3 are...

Recently we have been making studies [1, 2] on the Fibonacci number system that appears to be of considerable importance to the Fibonacci computer [3], A specific result that appears to be of particular interest is that through multilevel coefficients on a Fibonacci radix system, efficient extension of representations can occur. For example, through a ternary coefficient system, a doubling of t...

We use automata-theoretic approach to analyze properties of Fibonacci words. The directed acyclic subword graph (dawg) is a useful deterministic automaton accepting all suffixes of the word. We show that dawg’s of Fibonacci words have particularly simple structure. Our main result is a unifying framework for a large collection of relatively simple properties of Fibonacci words. The simple struc...

In [3], H. Belbachir and F. Bencherif generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. They prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers satisfying remarkable recurrence relations. [7], Mario Catalani define generalized bivariate polynomials, from which specifying initial conditi...