Quick observation: I recently noticed that Chebyshev polynomials and Fibonacci numbers have analogous formulas.
The nth Chebyshev polynomial satisfies for |x| ≥ 1, and the nth Fibonacci number is given by There’s probably a way to explain the similarity in terms of the recurrence relations that both sequences satisfy.
More on Chebyshev polynomials Yogi Berra and Chebyshev Product of Chebyshev polynomials The Brothers Markov More on Fibonacci numbers Fibonacci formula for pi Inverse Fibonacci numbers Fibonacci numbers and numerical integration.