Orthorecursive expansion of unity

We study the properties of a sequence [Formula: see text] defined by the recursive relation [Formula: see text] for [Formula: see text] and [Formula: see text]. This sequence also has an alternative definition in terms of certain norm minimization in the space [Formula: see text]. We prove estimates on the growth order of [Formula: see text] and the sequence of its partial sums, infinite series identities, connecting [Formula: see text] with the harmonic numbers [Formula: see text] and also formulate some conjectures based on numerical computations.