Packing parabola with the chain of internally tangent circles

Question

Related to the question Condition for perfect packing of ellipse with circles along the major axis

An attempt of "perfectly pack" a parabola $x=ay^2$ as a special king of ellipse with eccentricity 1 results in a packed sequence of circles

\begin{align} r_0&=\frac1{2a} ,\quad r_1=3r_0 ,\quad r_2=5r_0 ,\quad \dots ,\quad r_n=(2n+1)r_0 \tag{1}\label{1} , \end{align}

where $r_0$ is the starting circle, internally tangent to parabola at its vertex, which agrees with the curvature at the vertex.

Question: Is there any known reference where this property of parabola is mentioned/discussed?

Answer 1

There is a paper that came out in 2019: Giovanni Lucca, Integer sequences, Pythagorean triplets and circle chains inscribed inside a parabola. The PDF is here.

Abstract: In this paper we consider the infinite chains of mutually tangent circles that can be inscribed inside a parabola and we derive the expressions for the radii and centres coordinates; moreover, we establish the conditions that relate the circle chains to Pythagorean triplets and to certain integer sequences.