Asymptotically optimal k-step nilpotency of quadratic algebras and the Fibonacci numbers

It follows from the Golod–Shafarevich theorem that if $k\in{\mathbb N}$ and $R$ is an associative algebra given by $n$ generators and $d<n^2\cos^{-2}(\pi/(k+1))/4$ quadratic relations, then $R$ is not $k$-step nilpotent. We show that the above estimate is asymptotically optimal. Namely, for every $k$ there is a sequence of algebras $R_n$ given by $n$ generators and $d_n$ quadraticrelations such that $R_n$ is k-step nilpotent and $\lim_{n\to\infty d_n/n^2=\cos^{-2}(\pi/(k+1))/4$.