Computing a bit of Pi with the Lucas sequence

Here I have the sums of (1-tanh(n*a))^2, where a is log(phi) and n is the X axis.  So this is really the sums of Tanh', the first derivative.  Since the Lucas polynomials are Cyclotomic, they have roots on the unit circle.  These sums might approach the value Pi/2. They do, and get closest at the Lucas prime 29. After that, the sums stay close to Pi/2.

Here are the sums from my spread sheet:

0.8
1.2444444444
1.4444444444
1.5260770975
1.5580770975
1.5704227765
1.5751565043
1.5769672784
1.57765932
1.5779237128
1.5780247102
1.578063289
1.5780780249

Should we care? I am not sure. Whatever the starting angle, the series sums converge to some number since Tanh goes to 1.0.  So I have to show that somehow the series sums from ln(phi) converge to a specific value of pi. But Professor Lucas may have already figured that out. I am not surprised that it might, I just want to know if this is a relatively unique series from the hyperbolic angles made of phi.

The sinh and cosh still obey this:

 So any derivation of Pi from Pythagorean can be derived from this. The Taylor series of Tanh is limited to Pi/2 because this is a triangle.  Why Phi work likely goes back to Lagrange.

But leave that to later.  I am more interested in the differential:
sum(tanh'(n*a)) = pi/2. The residual error on that series is (tanh')^n when the series has n-1 terms. N is about 7.  That is a large power. It mean that light ultimately has 7 degrees of freedom, or there abouts. That also implies a big charge of six, I think, in the gluons. But I had estimated about 4, so dunno?

But this point conforms with what the physicists are doing with the natural units, they are making pi and output, not a constant. It seems all those tiny constituents of the vacuum seem concerned about getting an accurate value for pi. It is back to sphere packing I presume.