More, If you may recall the problem http://www.infoarena.ro/problema/kfib.
One of the solutions was with the golden ratio and if you never knew why and how those numbers were chosen, here is explained the solution of kth fibonacci number with eigenvectors, https://medium.com/@andrew.chamberlain/the-linear-algebra-view-of-the-fibonacci-sequence-4e81f78935a3.