Related papers: Counting base phi representations

In this work, we established symmetric representation of numbers where one can use any of 9 digits giving the same number. The representations of natural numbers from 0 to 1000 are given using only single digit in all the nine cases, i.e.,…

We consider the class numbers of imaginary quadratic extensions $F(\sqrt{-p})$, for certain primes $p$, of totally real quadratic fields $F$ which have class number one. Using seminal work of Shintani, we obtain two elementary class number…

Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.

In this paper, we give a new representation of the Fibonacci numbers. This is achieved using Fibonacci trees. With the help of this representation, the nth Fibonacci number can be calculated without having any knowledge about the previous…

Using combinatorial techniques, we derive a recurrence identity that expresses an exponential power sum with negative powers in terms of another exponential power sum with positive powers. Consequently, we derive a formula for the power sum…

We introduce two families of transcendental numbers which we call finite factorial (FF) and partially finite factorial (PFF) numbers respectively, with the former one being subfamily of the latter one. These numbers arise naturally from…

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…

For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We…

Let $q$ be a power of a prime $p$, $G$ be a finite abelian group, where $p$ does not divide $|G|$,and let $n$ be a positive integer. In this paper we find a formula for the number of irreducible representations of $G$ of a given dimension…

Throughout history, recreational mathematics has always played a prominent role in advancing research. Following in this tradition, in this paper we extend some recent work with crazy sequential representations of numbers- equations made of…

Let $F_n(k)$ be the generalized Fibonacci number defined by (with $F_i(k)$ abbreviated to $F_i$): $F_n = F_{n-1} + F_{n-2} + \dots + F_{n-k}$, for $n \geq k$, and the initial values $(F_0,F_1,...,F_{k-1})$. Let $B_n(k,j)$ be $F_n(k)$ with…

Let $B$ be an infinite subset of $\mathbf{N}$. When we consider partitions of natural numbers into elements of $B$, a partition number without a restriction of the number of equal parts can be expressed by partition numbers with a…

The following magic trick is at the center of this paper. While the audience writes the first ten terms of a Fibonacci-like sequence (the sequence following the same recursion as the Fibonacci sequence), the magician calculates the sum of…

In this article we prove that equation $\phi(x)=n$, for a fixed $n$, admits a finite number of solutions, we find the general form of these solutions, and we show that: if $x_0$ is a unique solution of this equation then $x_0$ is a product…

Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's natural to ask how many summands are needed. Using a continued fraction approach,…

A set A=A_{k,n} in [n]\cup{0} is said to be an additive k-basis if each element in {0,1,...,kn} can be written as a k-sum of elements of A in at least one way. Seeking multiple representations as k-sums, and given any function phi(n), with…

The symbolic representation of a number should be considered as a data structure, and the choice of data structure depends on the arithmetic operations that are to be performed. Numbers are almost universally represented using position…

Fibonacci polynomials are generalizations of Fibonacci numbers, so it is natural to consider polynomial versions of the various results for Fibonacci numbers. According to Hong, Pongsriiam, Bulawa, and Lee, the generating function of the…

We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer…

Let $\psi$ and $F$ be positive definite forms with integral coefficients of equal degree. Using the circle method, we establish an asymptotic formula for the number of identical representations of $\psi$ by $F$, provided $\psi$ is…