5 problems

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 $\mathcal{T}_P$ denote the Thue–Morse sequence along a polynomial subsequence, where $P$ is a polynomial of degree $d$. Write $M(\mathcal{T}_P,N)$ for its maximum order complex…

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…

Let ${\cal T}'=(t_{n^2})_{n\geq 0}$ be the subsequence of the Thue–Morse sequence along the squares, and let $M({\cal T}',N)$ denote its $N$th maximum order complexity. The notatio…