The Sum of the k-th Summands for the Jacobsthal Compositions of a Positive Integer

摘要

For a given positive integer n, let a_n count the number of compositions of n where the last summand is odd. Then, for n > 3, we have a_n = a_(n-1)(for when '1+' is appended to the front of the compositions of n - 1) + a_(n-2) (for when '2+' is appended to the front of the compositions of n - 2) + a_(n_2) (for when 2 is added to the first summand of each of the compositions of n - 2). The solution for the recurrence relation a_n = a_(n-1) + 2 a_(n-2), a_1 = 1, a_2 = 1 is 1/3(-1)~(n-1) + 2/3(2~(n-1)), n > 1. Since a_(n+1) = J_n, the n-th Jacobsthal number, these compositions are sometimes referred to as the Jacobsthal compositions of n. In this paper we determine s(n, k), the sum of all the k-th summands of the a_n Jacobsthal compositions of n, for n > k > 1, as well as several properties exhibited by these numbers.