Abstract
Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with n≥2, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n. Similar expressions are also dealt with for the case of Lucas numbers. Symmetric and anti-symmetric properties of Fibonacci and Lucas numbers are used in the proofs.
Highlights
Let Fn denote the nth Fibonacci number defined byF0 = 0, F1 = 1, Fn = Fn−1 + Fn−2 ( n ≥ 2).Lucas numbers Ln are defined as L0 = 2, L1 = 1, and Ln = Ln−1 + Ln−2 for n ≥ 2
One can find a lot of properties about Fibonacci and Lucas numbers in any book of Fibonacci numbers
Fibonacci numbers of negative indices can be defined in a natural way and Binet’s formula holds true for all n ∈ Z
Summary
Fibonacci numbers of negative indices can be defined in a natural way and Binet’s formula holds true for all n ∈ Z. Every positive integer can be expressed as a sum of nonconsecutive Lucas numbers of nonnegative indices. Α j Fj j=− N with each α j in {0, 1} and α j = 0 when j is a multiple of 3 No solutions of this problem were received at that time and so it was recently presented again by the editor Harris Kwong of the Fibonacci. We show in Theorems 2, 3 and 6 that, given any integers n and r with n ≥ 2, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n. Since the methods of the proofs in this article are constructive, it would be good if one may exploit our representations to construct new codes
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