Compositions and the Alternate Fibonacci Numbers

摘要

For a given positive integer n, a composition of n is an ordered sum of positive integers that sum to n. For example, 1 + 1 + 1, 1 + 2, 2 + 1, and 3 are the four compositions of 3. If, however, we have two different kinds of ones - say 1 and 1', then we find that there are 13 compositions of 3. We'll show that for the positive integer n, when we have two different kinds of ones, there are F_(2n+1) compositions of n. [F_n denotes the n-th Fibonacci number, where F_0 = 0, F_1 = 1, and F_n = F_(n-1) + F_(n-2), for n ≥ 2.] Then we'll determine the number of times a positive integer k ( ≤ n) appears as a summand among these F_(2n+1) compositions of n. This will be followed by determining the numbers of plus signs and summands for these F_(2n+1) compositions of n, and then the numbers of odd and even summands. Next we'll find the numbers of rises, levels, and descents for the compositions and, finally, the number of palindromes among the compositions will be determined.