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Zeckendorf's Theorem states that every positive integer can be uniquelyrepresented as a sum of non-adjacent Fibonacci numbers, indexed from $1, 2, 3,5,\ldots$. This has been generalized by many authors, in particular to constantcoefficient fixed depth linear recurrences with positive (or in some casesnon-negative) coefficients. In this work we extend this result to a recurrencewith non-constant coefficients, $a_{n+1} = n a_{n} + a_{n-1}$. Thedecomposition law becomes every $m$ has a unique decomposition as $\sum s_ia_i$ with $s_i \le i$, where if $s_i = i$ then $s_{i-1} = 0$. Similar toZeckendorf's original proof, we use the greedy algorithm. We show that almostall the gaps between summands, as $n$ approaches infinity, are of length zero,and give a heuristic that the distribution of the number of summands tends to aGaussian. Furthermore, we build a game based upon this recurrence relation,generalizing a game on the Fibonacci numbers. Given a fixed integer $n$ and aninitial decomposition of $n= na_1$, the players alternate by using movesrelated to the recurrence relation, and whoever moves last wins. We show thatthe game is finite and ends at the unique decomposition of $n$, and that eitherplayer can win in a two-player game. We find the strategy to attain theshortest game possible, and the length of this shortest game. Then we show thatin this generalized game when there are more than three players, no player hasthe winning strategy. Lastly, we demonstrate how one player in the two-playergame can force the game to progress to their advantage.

Extending Zeckendorf's Theorem to a Non-constant Recurrence and the Zeckendorf Game on this Non-constant Recurrence Relation