# On a mysterious recurrent sequence

Consider a simple minimization process: we start with a real polynomial $p_0 = 1$ and for $n > 0$ we define $p_{n+1} = p_n + c_{n+1} x^{n+1}$, where $c_{n+1} \in \mathbb{R}$ chosen in such a way that the L^2-norm of $p_{n+1}$ on [0,1] is minimized. At first we were wondering whether $||p_n||_2 \rightarrow 0$, but by the time we established that the limit is not zero, we had realized that the sequence of the coefficients $(c_n)$ is far more interesting to look at.

This is a sequence of rational numbers which satisfies a simple recurrence relation: $c_0 = 1$, and for $n > 0$ we have $c_0/(n+1) + \dots + c_n/(2n+1) = 0$, but not much is known about this sequence. In my talk I will present the results of a project (https://arxiv.org/abs/1901.04044), dedicated to studying the sequence. In particular, I will talk about:

1)estimates on the growth order of $c_n$ and the sequence of its partial sums,

2)a connection between $c_n$ and the harmonic numbers $H_n$,

and also I will formulate some conjectures based on numerical computations. For example, we conjecture that $c_n$ changes sign infinitely often, but we haven't obtained a proof of this fact!