Search engine for discovering works of Art, research articles, and books related to Art and Culture
Javascript must be enabled to continue!
Around the Fibonacci Numeration System
View through CrossRef
Let 1, 2, 3, 5, 8, … denote the Fibonacci sequence beginning with 1 and 2, and then setting each subsequent number to the sum of the two previous ones. Every positive integer n can be expressed as a sum of distinct Fibonacci numbers in one or more ways. Setting R(n) to be the number of ways n can be written as a sum of distinct Fibonacci numbers, we exhibit certain regularity properties of R(n), one of which is connected to the Euler φ-function. In addition, using a theorem of Fine and Wilf, we give a formula for R(n) in terms of binomial coefficients modulo two.
Title: Around the Fibonacci Numeration System
Description:
Let 1, 2, 3, 5, 8, … denote the Fibonacci sequence beginning with 1 and 2, and then setting each subsequent number to the sum of the two previous ones.
Every positive integer n can be expressed as a sum of distinct Fibonacci numbers in one or more ways.
Setting R(n) to be the number of ways n can be written as a sum of distinct Fibonacci numbers, we exhibit certain regularity properties of R(n), one of which is connected to the Euler φ-function.
In addition, using a theorem of Fine and Wilf, we give a formula for R(n) in terms of binomial coefficients modulo two.
Related Results
Some Properties of the Fibonacci Sequence
Some Properties of the Fibonacci Sequence
The purposes of this paper are; (a) to develop a relationship between subscripts of the symbols of Fibonacci and Lucas numbers and the numbers themselves; (b) to develop relationsh...
Beauty of Plants and Flowers Obeys Fibonacci Sequences
Beauty of Plants and Flowers Obeys Fibonacci Sequences
The aim of the study is to test the hypothesis that plants or flowers exhibiting Fibonacci sequences with larger numbers possess greater aesthetic beauty. As the number of Fibonacc...
Exploring fibonacci cordiality in corona graphs
Exploring fibonacci cordiality in corona graphs
A Fibonacci cordial (FC) labeling of a graph G is an injective function f : V(G) → {F0, F1, …, Fn}, where Fi is the ith Fibonacci number, such that the induced edge labeling f*(uv)...
On Matrices with Bidimensional Fibonacci Numbers
On Matrices with Bidimensional Fibonacci Numbers
Abstract
In this paper, the bidimensional extensions of the Fibonacci numbers are explored, along with a detailed examination of their properties, characteristics, a...
Pauli Gaussian Fibonacci and Pauli Gaussian Lucas Quaternions
Pauli Gaussian Fibonacci and Pauli Gaussian Lucas Quaternions
We have investigated new Pauli Fibonacci and Pauli Lucas quaternions by taking the components of these quaternions as Gaussian Fibonacci and Gaussian Lucas numbers, respectively. W...
Fibonacci Type Semigroups
Fibonacci Type Semigroups
We study “Fibonacci type” groups and semigroups. By establishing asphericity of their presentations we show that many of the groups are infinite. We combine this with Adjan graph t...
Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers
Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers
AbstractLet (Fn)n≥0and (Ln)n≥0be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and...
Representation of Integers as Sums of Fibonacci and Lucas Numbers
Representation of Integers as Sums of Fibonacci and Lucas Numbers
Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with n≥2, every positive integer can be expressed as a sum of Fibonac...