Fibonacci Numbers and Their Lucas Coefficients
Number Theory
2025-09-03 v1
Abstract
We show that for the classical Fibonacci sequence (Fn) and the Lucas sequence (Ln) the following identity holds for every integer n >= 2: (n-1)Fn equals the sum from k=1 to n-1 of Lk multiplied by F(n-k). Equivalently, this gives a representation of the nth Fibonacci number as Fn = (1 / (n-1)) times the same sum. We present a detailed proof by mathematical induction and illustrate the identity with a numeric example.
Cite
@article{arxiv.2509.00070,
title = {Fibonacci Numbers and Their Lucas Coefficients},
author = {Tapan Suthar},
journal= {arXiv preprint arXiv:2509.00070},
year = {2025}
}
Comments
5 pages, elementary note on Fibonacci Lucas identity