Compositions with the Last Summand Odd

Abstract

For \(n \geq 1\), we let \(a_n\) count the number of compositions of the positive integer \(m\), where the last summand is odd. We find that \(a_n = (\frac{1}{3})(-1)^n + (\frac{2}{3}) 2^{n-1}\). Since \(J_n\), the \(n\)-th Jacobsthal number, is given as \(\frac{1}{3}(-1)^n + \frac{2}{3}2^{n-1}\) for \(n \geq 0\), it follows that \(a_n = J_{n-1}\) for \(n \geq 1\). For this reason, these compositions are often referred to as the Jacobsthal compositions.

In our investigation, we determine results for the \(a_n\) compositions of \(n\), such as: (i) \(a_{n,k}\), the number of times the positive integer \(k\) appears as a summand among these \(a_n\) compositions of \(n\); (ii) the numbers of plus signs, summands, even summands, and odd summands that occur for these compositions of \(n\); (iii) the sum of the even summands and the sum of the odd summands for the \(a_n\) compositions of \(n\); (iv) the numbers of levels, rises, and descents for the \(a_n\) compositions; and (v) the number of runs that occur among these \(a_n\) compositions.