Stern’s diatomic sequence and a series of tangent circles orthogonal to the unit circle

We study a series of tangent circles orthogonal to the unit circle on the complex plane. In particular, we study the case that the number of tangent circles is three. By operat-ing inversions to the circles, we have an infinite family of circles. We show that the inverse of the radius of a circle in the family is a linear sum of the inverses of the radii of beginning three circles, and then their coefficients are expressed by using Stern’s diatomic sequence (Theorem 4.4). As a corollary, we obtain a formula to compute π (Corollary 4.5).