9 problems

Let $k\geq 3$ and let $\vec{\mathbf{v}}\in\mathbb{Z}^{k-1}\setminus\{\vec{\mathbf{0}}\}$. In the Large Steps Algorithm, let $j_{\mathrm{lsb}}$ denote the bound on the largest index…

Let $k$ be the length of the vector $\vec{\mathbf{c}}=(c_1,\ldots,c_k)$, and let the algorithm in the proof of Theorem main be the conversion algorithm for $\vec{\mathbf{c}}$-repre…

Let the forward and reverse Zeckendorf games be played from corresponding starting positions, with each game having a winning player under optimal play. Forward–reverse Zeckendorf…

For a sequence $\mathcal{S}$, let a step be an integer $N$ at which its maximum order complexity changes from $M(\mathcal{S},N-1)$ to $M(\mathcal{S},N)$, and define the ratio of su…

Let $s_Z(n)$ denote the sum of digits of $n$ in Zeckendorf base, let $\mathcal{S}_Z=(s_Z(n)\bmod 2)_n$, and let $\mathcal{S}_{Z,P}$ be the sequence obtained by restricting…

Let $\varphi=(1+\sqrt{5})/2$ be the golden ratio, let $s_Z(n)$ denote the sum of digits of $n$ in Zeckendorf base, and let $\mathcal{S}_Z=(s_Z(n)\bmod 2)_n$. Write…

Consider the Tribonacci Game played for an integer $n$, with Player 1 and Player 2 taking turns according to the game's rules. Player 2's eventual winning-strategy conjecture. Play…

Let the Tribonacci game be the game on representations of an integer in which legal moves are those prescribed by the Tribonacci recurrence, and let the deterministic game be the g…

For a uniformly random Zeckendorf Game on $n$ initial copies of $1$, let $M_n$ be the number of moves until the Zeckendorf decomposition of $n$ is reached, and let…