The Fibonacci Diatomic Array Applied to Fibonacci Representations

Abstract

The Fibonacci diatomic array of this paper is a self-generating array which begins with two rows, each containing two 1’s; each row begins and ends with two 1’s. Each interior element in each row is in row (n - 1) or else is the sum of two adjacent elements in row (n - 2), according as the column number is of the form a _p or b _p , where (a _p , b _p ) is a Wythoff pair. The Fibonacci diatomic array counts the number of Fibonacci representations R(N) of non-negative integers N. It is also possible to generate the Fibonacci diatomic array from a single row of two 1’s, or to use a single row of two 1’s to generate either the odd rows or the even rows of the array. Both the array of Fibonacci representations and the Fibonacci diatomic array of this paper illustrate many known identities relating to Fibonacci representations while suggesting new identities.