GENERALIZED PADOVAN AND POLYNOMIAL SEQUENCES

Copyright © 2021 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: By generalizing the well-known classical Padovan Sequence using Quadratic, Cubic and general polynomial sequences as coefficients we arrive at various limiting ratios in this paper. Few illustrations are provided to justify the results arrived regarding the limiting ratios. The idea of limiting ratio will provide the asymptotic behavior of the terms of the sequence. In this sense, the results obtained in this paper will provide more information about the behavior of Padovan sequence in combination with general polynomial sequences.


INTRODUCTION
The classic Padovan sequence was named after British polymath Richard Padovan and it was subsequently popularized by Ian Stewart through his books. It is well known that the ratio of successive terms of Padovan sequence approach a number called Plastic number given by 1.32471 approximately. In this paper, by considering quadratic, cubic and general polynomial sequences we will construct certain recurrence relations and obtain interesting results regarding 220 R. SIVARAMAN the limiting ratios of such relations. This will provide a new insight of the behavior of class of Padovan sequences in more general form.

LIMITING RATIO
The ratio of (n + 1)th term to the nth term of any sequence as n →is defined as Limiting Ratio of that sequence (2.1). We denote the limiting ratio by  . , , , P Q R S are some real numbers such that P is non-zero. kA + (5.5). Note that this value doesn't depend of the constants B and C.

5.1.1
Let us consider central polygonal numbers whose kth term is given by . The terms of this sequence are 1, 2, 4, 7, 11, 16, 22, 29, . . . We observe that these numbers are one plus the triangular numbers. These numbers describe the maximum number of pieces that can be made in a pancake or pizza (or any circular object) with a given number of straight cuts. In this sense, this sequence is also sometimes informally referred as Lazy caterer's sequence.
By definition (3.1), we notice that the Lazy caterer's sequence is a Quadratic sequence with 11 , , 1 22 If we now consider the generalized Padovan sequence along with Lazy caterer's sequence as defined in (5.1) then from (5.3) we find that the limiting ratio  is the positive root of approximately. We notice that this value agree with the actual computation up to first three decimal places. We also notice that for small values of A, the limiting ratio is very close to 2 k (5.9). In our case since, k = 10 and A = 0.5 is very small, the limiting ratio is very close to 2 100 k = .

GENERALIZED PADOVAN AND CUBIC SEQUENCES
Let (1)  Hence they are also called as Triangular Pyramidal Numbers.
By definition (4.1), we notice that Tetrahedral numbers sequence is a cubic sequence with approximately. We notice that this value agree with the actual computation up to first two decimal places. We also notice that for small values of P, the limiting ratio is very close to 3 k (6.9). In our case since, k = 5 and P = 0.1666 is very small, the limiting ratio is very close to 3 125 k = . Thus, as k →, the limiting ratio of generalized Padovan sequence and general polynomial sequence as defined in (7.1) is 2 0 n ka + (7.5). The result obtained in (7.5) gives the more general case and we notice that the limiting ratios depends only on k and 0 a . We also notice that if the constant term 0 a of the polynomial sequence is small then from (7.5), we see that the limiting ratio will be very close to n k .

CONCLUSION
By considering Quadratic, Cubic and General polynomial sequence of nth degree, and constructing specific recurrence relations for each of them given by equations (5.1), (6.1), (7.1) we have obtained limiting ratios provided by the equations (5.5), (6.5), (7.5) respectively. The limiting ratios obtained for Quadratic and Cubic sequences are verified by two suitable illustrations in each case. In all the cases we see that if the constant term of the polynomial sequence of nth degree is a small number then the limiting ratio is approximately n k giving a nice relationship between the degree of the polynomial sequence and the limiting ratio. This paper thus establishes the existence of limiting ratios of general Padovan sequence combined with general polynomial sequences of nth degree.