Writing π as sum of arctangents with linear recurrent sequences, Golden mean and Lucas numbers

Abstract

In this paper, we study the representation of π as sum of arcotangents. In particular, we obtain new identities by using linear recurrent sequences. Moreover, we provide a method in order to express π as sum of arcotangents involving the Golden mean, the Lucas numbers, and more in general any quadratic irrationality.A(x) = arctan(x).It is well-known that for x, y ≥ 0, if y =

Finding golden mean in a physics exercise

“…Probably, the most famous number in the world is π, the ratio of a circle's circumference to its diameter, but another number, known since antiquity, seems to be a part of the intimate structure of the Universe [1]. It is usually denoted by ϕ, perhaps in honour of the famous Greek sculptor Fidia.…”