Related papers: Fibonacci and digit-by-digit computation; An examp…
The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known and algorithms to compute them are so easy that they are often used…
We compute the Frobenius number for numerical semigroups generated by the squares of three consecutive Fibonacci numbers. We achieve this by using and comparing three distinct algorithmic approaches: those developed by Ram\'irez Alfons\'in…
The following article summarizes research where theorems and their respective demonstrations are postulated based on quadratic equations with special properties given by the Pythagorean triplets and the Fibonacci sequence given the second…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
In this paper, we present a new approach to the convolved Fibonacci numbers arising from the generating function of them and give some new and explicit identities for the convolved Fibonacci numbers.
Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$, we introduce the Fibonacci partition as a Fibonacci permutation of its blocks. Then we define…
The focus of this note is to formulate the algorithms and give the examples used by Fibonacci in Liber Abaci to expand any fraction into a sum of unit fractions. The description in Liber Abaci is all verbal and the examples are numbers…
By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph…
The Fibonacci sequence $\mathbb{F}$ is the fixed point beginning with $a$ of morphism $\sigma(a,b)=(ab,a)$. In this paper, we get the explicit expressions of all squares and cubes, then we determine the number of distinct squares and cubes…
In this article, we present a trick around Fibonacci numbers which can be found in several magic books. It consists in computing quickly the sum of the successive terms of a Fibonacci-like sequence. We give explanations and extensions of…
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…
A generalization of the well--known Fibonacci sequence is the $k$--Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,\ldots,0,1$, and each term afterwards is the sum of the preceding $k$ terms.…
We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.
The Fibonacci sequence is a sequence of numbers that has been studied for hundreds of years. In this paper, we introduce the new sequence S_{k,n} with initial conditions S_{k,0} = 2b and S_{k,1} = bk + a, which is generated by the…
The classical Fibonacci sequence is known to exhibit many fascinating properties. In this paper, we explore the Fibonacci sequence and integer sequences generated by second order linear recurrence relations with positive integer…
One possible data encryption scheme is related to stream ciphers, which use a sufficiently long pseudo-random sequence. To increase the cryptographic strength of the cipher, linear shift algorithms (generated by linear recurrent sequences…
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…
Based on the structure of Fibonacci sequence, we give a new proof for the irrationality exponents of the Fibonacci real numbers. Moreover, we obtain all the irrationality exponents of the real numbers corresponding to the differences of…
An interesting open problem in number theory asks whether it is possible to walk to infinity on primes, where each term in the sequence has one more digit than the previous. In this paper, we study its variation where we walk on the…
Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. For all integers $a$ and $b \geq 1$ with $\gcd(a, b) = 1$, let $[a^{-1} \!\bmod b]$ be the multiplicative inverse of $a$ modulo $b$, which we pick in the usual set of…