Abstract
A sequence of four compositions of 3 is: 1 + 1 + 1, 1 + 2, 2 + 1, 3. By the replacement of the plus signs (+) and commas (,) by the multiplication dots (?) and plus signs (+) respectively, the sequence becomes the summation series: 1?1?1 + 1?2 + 2?1 + 3, which is equal to 8 or 6 th number in the famous Fibonacci sequence. It is a curious fact that the sum of a positive integer n and the products of summands corresponding to the compositions of n is equal to (2n)-th Fibonacci number. We establish the proposition after obtaining a special order of the compositions of n; and then obtain some other results. The results show that Fibonacci sequence has close connection with the special order of the compositions of n. Two Fibonacci identities, which we derive from a special recurrence relation, are useful to prove two theorems. The relationships are stated first in the theorems and are then shown in the consequences of the theorems.
Highlights
Partitions of a positive integer n including permutations of the parts or summands are called compositions of n
It is a curious fact that the sum of a positive integer n and the products of summands corresponding to the compositions of n is equal to (2n)-th Fibonacci number
The results show that Fibonacci sequence has close connection with the special order of the compositions of n
Summary
Partitions of a positive integer n including permutations of the parts or summands are called compositions of n. Eight compositions of 4 are: 4, 3 + 1, 1 + 3, 2 + 2, 2 + 1 + 1, 1 + 2 + 1, 1 + 1 + 2, 1 + 1 + 1 + 1. Fibonacci sequence is the following sequence of integers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,. There exists a definite order of the compositions of n, which has close connection with the Fibonacci sequence. We use the following simple notations for compositions of n to establish the relationship. + xr = n denotes some C(n) under which the compositions start with the common summands: x1, ... We use the symbol of equivalence (≡) between {C(n)} and its implication; and between x1 + ...
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