Padovan numbers as sums over partitions into odd parts

Cristina Ballantine,Mircea Merca

doi:10.1186/s13660-015-0952-5

Abstract

Recently it was shown that the Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. In this paper, we introduce a similar representation for the Padovan numbers. As a corollary, we derive an infinite family of double inequalities.

Highlights

Integer sequences appear often in many branches of science
Merca [ ], Corollary, proved that the Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts, i.e., Fn =
5 Results and discussion According to Theorem, the nth Padovan number can be expressed as a sum of multinomial coefficients over integer partitions of n into odd parts greater than

Summary

Introduction

Integer sequences appear often in many branches of science. One famous example is the Fibonacci numbers that have been known for more than two thousand years and find applications in mathematics, biology, economics, computer science, physics, engineering, architecture, and so forth [ , ]. Merca [ ], Corollary , proved that the Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts, i.e., Fn = The nth Padovan number can be expressed in a similar way as a sum of multinomial coefficients over integer partitions of n into odd parts.

Results

Conclusion