Completeness of Positive Linear Recurrence Sequences

Abstract

Zeckendorf proved that every positive integer n can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this to create a two-player game. Given a fixed integer n and an initial decomposition of $n = n F_1$, the two players alternate by using moves related to the recurrence relation $F_{n+1} = F_n + F_{n-1}$, and whoever moves last wins. The game always terminates in the Zeckendorf decomposition, though depending on the choice of moves the length of the game and the winner can vary. We find upper and lower bounds on the number of moves possible. The upper bound is on the order of $n\log n$, and the lower bound is sharp at $n-Z(n)$ moves, where Z(n) is the number of terms in the Zeckendorf decomposition of n. Notably, Player 2 has the winning strategy for all $n > 2$; interestingly, however, the proof is non-constructive.

The authors were partially supported by NSF grants DMS1265673 and DMS1561945, the Claire Booth Luce Foundation, and Carnegie Mellon University. We thank the students from the Math 21–499 Spring ’16 research class at Carnegie Mellon and the participants from CANT 2016 and 2017 and the 18th Fibonacci Conference, especially Russell Hendel, for many helpful conversations.