We study B(n;k), the number of ways of writing n as a sum or difference of the first k Fibonacci numbers. We show that B(0;k) satisfies the Tribonacci-like recurrence B(0;k+1)=B(0;k)+B(0;k−1)+B(0;k−2) and that B(n;k) satisfies a modified version of this recurrence.
@article{arxiv.2604.15446,
title = {Non-standard Zeckendorf decompositions; or, Tribonacci within Fibonacci},
author = {Katie Anders and Madeline L. Dawsey and Joseph Vandehey},
journal= {arXiv preprint arXiv:2604.15446},
year = {2026}
}