Unique Representation of Positive Integers as a Sum of Distinct Tribonacci Numbers

Corresponding Author: Salim Badidja Laboratory of Pure and Applied Mathematics (L.M.P.A.), Faculty of Mathematics and Computer Sciences, University of Mohamed BOUDIAF of M’sila, M’sila, Algeria Email: bsalimmah@gmail.com Abstract: Let (Tm)m≥1 be the tribonacci sequence. We show that every integer N ≥ 1 can be written as a sum of the terms αm Tm, where m runs over the set of strictly positive integers and αm (m ≥ 1) are either 1 or 0. The previous representation of N is unique if each time that we have αm = 1 then at least the two coefficients directly following αm are zero, i.e., αm+1 = αm+2 = 0.


Theorem 2.1
For every integer N > 0, there exists an integer i 1 > 0 such that

Proof
Let N > 0 be an integer. Firstly, we show that the sequence (T i ) i≥1 satisfies: Indeed, we easily check this for i = 1, 2, 3. For i≥ 4, we have by definition: Now, putting ∆ 0 = N. Let i 1 be the largest integer such that: Continuing in this process, we obtain a decreasing sequence i 1 > i 2 > i 3 >... > ... which should stabilize at some i k . Hence

Proposition 2.1
For every integer n ≥ 5, we have:

Proof
Before the proof we need the following lemma.

Lemma 2.1
Under the same assumption of theorem 2.2 and for k ≥ 6, we have: Firstly we prove by induction that: This formula is satisfied for k = 6, because: and prove that: Since T 2 -1 = 0 ≥ 0 and the fact that (T i ) i≥4 is strictly increasing, we have always: This completes the proof.
Returning to the proof of the lemma. Putting: and prove that L > 0. We distinguish two cases: A) For α k = 0, we have: and from above: That is, Let k be the largest value of m such that α k ≠ β k , we may assume without loss of generality that α k ≠ 1 and β k = 0. Since the validity of lemma 2.1 and proposition 2.1 is for k ≥ 6, we distinguish in the sequel two cases. A) 1 ≤ k ≤ 6.
With these coefficients we can represent N as: Finally, we have: which is a contradiction. Hence the representation is unique.

Applications and Perspectives
The representation of a positive integer n by a sum of elements of a given sequence is an interesting problem which is well known in the mathematical literature; namely, unique representation of integers as sum of distinct Lucas numbers (Brown, Jr, 1969), Fibonacci and Lucas representation (Pihko, 1986), Cantor's development of a positive integer (Mercier, 2004),... . Our problem set in this context but with a well addition which is the uniqueness of representation.
As a perspective, the techniques used in our work can be employed in other problems for the same purposes. In this sense we can consider, for example, the case of the generalized tribonacci sequences and higher orders (Pentanacci, hexanacci, ...k-Fibonacci sequence..).
This field of mathematics which focuses on the study of words and formal languages combinatorics on words affects various areas of mathematics study, including algebra and computer science. Combinatorics of words is connected to many modern, as well as classical, fields of mathematics. Connections to combinatorics-actually being part of it -are obvious, but also connections to algebra are deep. Indeed, a natural environment of a word is a free semigroup.
More generally, the above connections can be illustrated as in Fig. 1 (Karhumaki, J.).
For more clarification we can take the sequences of words like the Fibonacci sequence of words on the binary alphabet {0, 1} can be de.ned by the recurrent relation: F 1 = 1, F 2 = 0, F 3 = 01, F 4 = 010,.... . For further references on the subject see for example (Lotaire, 2002;Karhumaki and Berstel, 2003).