Properties of Padovan Sequence

Among several types of famous sequences that exist in all branches of mathematics, Padovan sequence is one of such sequences possessing amusing properties. In this paper, we shall discuss about Padovan Sequence and determine a special number called Plastic number. We also derive the ratio of its (n+1)st term to the nth term as n is very large, which is called “Limiting Ratio”. Finally we discuss the geometrical interpretation of the terms of Padovan sequence.


Introduction
Padovan Sequence is named after British Polymath Richard Padovan who became fascinated with the works of Dutch monk and architect Hans van der Laan to the extent that he wrote a book about him titled "Dom Hans van der Laan: modern primitive" in the year 1989. In this book Richard Padovan while mentioning works of Hans van der Laan attributed an interesting sequence of numbers which had striking resemblance with the most famous Fibonacci numbers. Later, British mathematician and science popularizer Ian Stewart while describing the same sequence in one of his books called the sequence as Padovan Sequence and the name remained thereafter. Though Padovan sequence resembles Fibonacci sequence, it differs quite significantly from Fibonacci numbers especially in the case of dealing with limiting ratio. We notice that the first three terms of the sequence are 1 and from the fourth term, each term is the sum of one but previous two terms. That is, fourth term is sum of second and first terms, fifth term is sum of third and second terms and so on. With this convention, the first few terms of the Padovan sequence (with help of (2.1)) are given by 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, . . . (2.2)

Characteristic Equation
From the recurrence relation used for generating Padovan sequence, we can obtain the Characteristic Equation for Padovan Sequence.
The auxiliary equation considering to this recurrence relation is Equation (3.1) is called Characteristic Equation of Padovan sequence. We will try to determine the real roots of the Characteristic equation.

Roots of the Characteristic Equation
Being a cubic equation in m, there is no straight forward method to determine the real roots of (3.1). First, we determine the number of real roots of the Characteristic equation.

Determining the Root
We try to determine that particular positive root. For this, we can consider Newton -Raphson Method.
If 0 x is the initial approximation lying in   (1) (2) ff  , let us assume initial approximation to the root to be 0 1 x = .
Using (3.2), we have The number 1.32471 which happens to be root of the characteristic equation pertaining to the recurrence relation of Padovan sequence is called "Plastic Number".

Graphical Verification
In Figure 1, we have graph of the function In this section, we try to obtain the ratio of (n+1)st term to the nth term of terms in the Padovan sequence as n →. We define such ratio as Limiting Ratio.
If we assume the limiting ratio as , then by definition, as n →,

 =+
But we see that equation (4.4) is exactly same as the characteristic equation that we obtained in (3.1). Hence, the only positive root of (4.4) is given by the plastic number We thus have proved that the limiting ratio of terms of Padovan sequence is the plastic number which is approximately 1.32471

Conclusion
One of the fascinating sequences in mathematics called Padovan sequence is introduced in this paper. Using the recurrence relation obtained from the definition of the sequence, we determined the characteristic equation whose only positive real root is calculated using Newton -Raphson method and found to be a real number 1.32471 approximately. This number is called Plastic Number. We proved that the limiting ratio, that is ratio of (n+1)st term to the nth term as n is very large, of terms of the Padovan sequence is the plastic number, the root of the characteristic equation. Thus, we see that the plastic number for Padovan sequence plays the same role as that of Golden Ratio does for Fibonacci sequence. We verified the calculation of plastic number using Figure 1 obtained using Desmos Online Graphing Calculator. We know that the Fibonacci numbers produces rectangles stacked upon continuously. Similarly when we try to visualize the terms of the Padovan sequence, we got spiraling equilateral triangles as shown in Figure 2. Thus, the Padovan sequence exhibits almost the same characteristics as that of its cousin sequence Fibonacci sequence, leading to a new limiting ratio called plastic number. We can easily generalize this sequence and study the behaviour of such generalized Padovan sequences.