Fibonacci Numeration System Research Articles

ГЕОМЕТРИЗАЦИЯ ОБОБЩЕННЫХ СИСТЕМ СЧИСЛЕНИЯ ФИБОНАЧЧИ И ЕЕ ПРИЛОЖЕНИЯ К ТЕОРИИ ЧИСЕЛ

Generalized Fibonacci numbers { F (g) i } are defined by the recurrence relation F (g) i+2 = gF(g) i+1 + F (g) i with the initial conditions F (g) 0 = 1, F (g) 1 = g. These numbers generater representations of natural numbers as a greedy expansions n = ∑k i=0 εi(n)F (g) i , with natural conditions on εi(n). In particular, when g = 1 we obtain the well-known Fibonacci numeration system. The expansions obtained by g > 1 are called representations of natural numbers in generalized Fibonacci numeration systems. This paper is devoted to studying the sets F (g) (ε0, . . . , εl), consisting of natural numbers with a fixed end of their representation in the generalized Fibonacci numeration system. The main result is a following geometrization theorem that describe the sets F (g) (ε0, . . . , εl) in terms of the fractional parts of the form {nτg}, τg = √ g 2+4−g 2 . More precisely, for any admissible ending (ε0, . . . , εl) there exist effectively computable a, b ∈ Z such that n ∈ F (g) (ε0, . . . , εl) if and only if the fractional part {(n + 1)τg} belongs to the segment [{−aτg}; {−bτg}]. Earlier, a similar theorem was proved by authors in the case of classical Fibonacci numeration system. As an application some analogues of classic number-theoretic problems for the sets F (g) (ε0, . . . , εl) are considered. In particular asymptotic formulaes for the quantity of numbers from considered sets belonging to a given arithmetic progression, for the number of primes from considered sets, for the number of representations of a natural number as a sum of a predetermined number of summands from considered sets, and for the number of solutions of Lagrange, Goldbach and Hua Loken problem in the numbers of from considered sets are established.

Searching for Zimin patterns

In the area of pattern avoidability the central role is played by special words called Zimin patterns. The symbols of these patterns are treated as variables and the rank of the pattern is its number of variables. Zimin type of a word x is introduced here as the maximum rank of a Zimin pattern matching x. We show how to compute Zimin type of a word on-line in linear time. Consequently we get a quadratic time, linear-space algorithm for searching Zimin patterns in words. Then we demonstrate how the Zimin type of the length n prefix of the infinite Fibonacci word is related to the representation of n in the Fibonacci numeration system. Using this relation, we prove that Zimin types of such prefixes and Zimin patterns inside them can be found in logarithmic time. Finally, we give some upper bounds on the function f(n,k) such that every k-ary word of length at least f(n,k) has a factor that matches the rank n Zimin pattern.

Geometrization of the Fibonacci numeration system, with applications to number theory

Geometrization of the Fibonacci numeration system, with applications to number theory

Complementary Iterated Floor Words and the Flora Game

Let $\varphi=(1+\sqrt{5})/2$ denote the golden section. We investigate relationships between unbounded iterations of the floor function applied to various combinations of $\varphi$ and $\varphi^2$. We use them to formulate an algebraic polynomial-time winning strategy for a new four-pile take-away game Flora, which is motivated by partitioning the set of games into subsets CompGames and PrimGames. We further formulate recursive, arithmetic, and word-mapping winning strategies for it. The arithmetic one is based on the Fibonacci numeration system. We further show how to generate the floor words induced by the iterations using word-mappings and characterize them using the Fibonacci numeration system. We also exhibit an infinite array of such sequences.

Fibonacci, Van der Corput and Riesz–Nágy

What do the three names in the title have in common? The purpose of this paper is to relate them in a new and, hopefully, interesting way. Starting with the Fibonacci numeration system — also known as Zeckendorff's system — we will pose ourselves the problem of extending it in a natural way to represent all real numbers in ( 0 , 1 ) . We will see that this natural extension leads to what is known as the ϕ-system restricted to the unit interval. The resulting complete system of numeration replicates the uniqueness of the binary system which, in our opinion, is responsible for the possibility of defining the Van der Corput sequence in ( 0 , 1 ) , a very special sequence which besides being uniformly distributed has one of the lowest discrepancy, a measure of the goodness of the uniformity. Lastly, combining the Fibonacci system and the binary in a very special way we will obtain a singular function, more specifically, the inverse of one of the family of Riesz–Nágy.

On representations of positive integers in the Fibonacci base

We exhibit and study various regularity properties of the sequence ( R ( n ) ) n ⩾ 1 which counts the number of different representations of the positive integer n in the Fibonacci numeration system. The regularity properties in question are observed by representing the sequence as a two-dimensional array consisting of an infinite number of rows L 1 , L 2 , L 3 , … where each L k contains f k - 1 (the k - 1 st Fibonacci number) entries of the sequence ( R ( n ) ) . We give a purely combinatorial recursive algorithm for generating each row L k from previous rows L j with j < k . We then show that for each positive integer m , and for all k ⩾ 2 m , the number of occurrences of m in L k is a constant rk ( m ) depending only on m . The function rk ( m ) has many interesting number theoretic properties and is intimately connected to the Euler φ -function.

Robust universal complete codes for transmission and compression

Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variable-length codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is constant and needs not be generated for every probability distribution. These codes can be used as alternatives to Huffman codes when the optimal compression of the latter is not required, and simplicity, faster processing and robustness are preferred. The codes are compared on several “real-life” examples.

Fibonacci representations and finite automata

Finite-state automata are used as a simple model of computation since only a finite memory is needed. The problem of passing from any representation to the normal representation of an integer within the Fibonacci numeration system, which is called the process of normalization, is addressed. It is shown that the normalization can be realized by means of infinite automata. More precisely, this function can be obtained by the composition of two subsequential transducers that are simply obtained from the linear recurrence definition of the basis of the Fibonacci system, one processing words from left to right and the other from right to left. The normalization, although not a sequential process, can be obtained in two sequential passes. It is proved that it is possible to add to integers written in the Fibonacci numeration system of order m by means of a finite-state automaton. The conversion from a Fibonacci representation to the standard binary representation cannot be realized by a finite-state automaton. >