An oddity from my spread sheet

I was working the problem of the hyperbolic Lucas solution when th second derivative of the polynomial laws of motion resulted in Phi*n, Phi^n n integer. Why? Because I cheat, I know that 4^Phi is 3*Pi, and the Phi is broken into a series sum of 1 + Phi^(-n).... So Phi*n appears as an exponent on Cosh in the solution.

So I get the sum  Ln^(phi^n), where the Ln are Lucas, meaning cosh. But I included sinh also. That is the Red line and it goes to 3, the yellow is the error and it smoothly declines. But computing the error for each of Nmax from 1 to 25 I find the minimal error at Lucas number 16.

This likely concerning the cyclochromatic zeros of the Lucas polynomials. The exponent here is filling in the angles about the circle, generally avoiding the zero; but making the circle. I might even take a closer look.