A Visual Tour of Identities for the Padovan Sequence

MacTutor History of Mathematics, 2014

Herein we investigate the historical origins of the Fibonacci numbers. After emphasising the importance of these numbers, we examine a standard conjecture concerning their origin only to demonstrate that it is not supported by historical chronology. Based on more recent findings, we propose instead an alternative conjecture through a close examination of the historical and historical/mathematical circumstances surrounding Leonardo Fibonacci and relate these circumstances to themes in medieval and ancient history. Cultural implications and historical threads of our conjecture are also examined in this light.

2022

In this paper we explore two types of tilings of a honeycomb strip and derive some closed form formulas for the number of tilings. Furthermore, we obtain some new identities involving tribonacci numbers, Padovan numbers and Narayana’s cow sequence and provide combinatorial proofs for several known identities about those numbers.

2004

Is there any other proportion for a rectangle, other than the Golden Proportion, that will allow the process of cutting off successive squares to produce an infinite paving of the original rectangle by squares of different sizes? The answer is: No. The only proportion that allows this pattern is the Golden Ratio. Two proofs are given.

2011

The Fibonacci numbers are numbers of the integer sequence 1, 1, 2, 3, 5, 8, 13, …, which are defined by a1= a2=1 and an+2 = an+1 + an, for n=1, 2, 3, …. By the history, the sequence of these numbers was first introduced officially by Leonardo Pisano Bigollo for solving the mathematical problem involving the growth of population of rabbits based on idealized assumptions. The objective of this research was to consider the other three mathematical problems which could be expressed by the Fibonacci numbers also. The problems were 1) determine the number of patterns for bricks rearran gement, 2) determine the number of subsets of {1,2,…, n} which do not contain two consecutive numbers, and 3) determine the number of paths for travelling between 5 cities suc h that the traveler can change direction at any intermediate city. The recurrence relation conc ept was used to show the connection between the problems and Fibonacci numbers. It showed that high school mathematics was able to explain...

Bulletin of the International Mathematical Virtual Institute, 2025

In this paper, we introduce the incomplete Padovan numbers. In addition we derive the recurrence relations, some properties of these numbers, and obtain the generating function of the incomplete Padovan numbers. Finally, we give a Python code for calculation of the incomplete Padovan numbers.

KoG, 2017

Fibonacci sequence and the limit of the quotient of adjacent Fibonacci numbers, namely the Golden Mean, belong to general knowledge of almost anybody, not only of mathematicians and geometers. There were several attempts to generalize these fundamental concepts which also found applications in art and architecture, as e.g. number series and quadratic equations leading to the so-called ``Metallic means'' by V. de Spinadel [8] or the cubic ``plastic number'' by van der Laan [5] resp. the ``cubi ratio'' by L. Rosenbusch [7]. The mentioned generalisations consider series of integers or real numbers. ``Non-standard aspects'' now mean generalisations with respect to a given number field or ring as well as visualisations of the resulting geometric objects. Another aspect concerns Fibonacci type resp. Padovan type combinations of given start objects. Here it turns out that the concept ``Golden Mean'' or ``van der Laan Mean'' also makes sense f...

Undergraduate Texts in Mathematics, 1997

MAA spectrum, 2006

Chaos Solitons & Fractals, 2007

The general k-Fibonacci sequence fF k;n g 1 n¼0 were found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition. This sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and related with the so-called Pascal 2-triangle.

Mathematics, 2021

A generalization of the well-known Fibonacci sequence is the k−Fibonacci sequence whose first k terms are 0,…,0,1 and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci numbers that are curious numbers (i.e., numbers whose base ten representation have the form a⋯ab⋯ba⋯a). This work continues and extends the prior result of Trojovský, who found all Fibonacci numbers with a prescribed block of digits, and the result of Alahmadi et al., who searched for k-Fibonacci numbers, which are concatenation of two repdigits.