Non-standard Zeckendorf decompositions; or, Tribonacci within Fibonacci

Katie Anders, Madeline L. Dawsey, Joseph Vandehey

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;…