Finite differences and recursions

Thi is pure applied math.  I am selecting a recursion of the form:
y^2 -a^y -1 = 0, which then generates a power series in 1+y+y^2+y^3.... and from that I compute tanh and the first two differences. Tanh is the blue and the red and yellow the first two differences.  I want to see how smoothly the selected ratio converges.

For example, I am starting with something simple, the fourth order Taylor expansion of a circle volume when the center is estimated. The I try out ratios that might accurately model that polynomial.  Physics has a lot of dimensionless ratios, 127,1836,119*pi are particular examples.  These ratios have more to do with the spectrum of a sphere then anything else, and ultimately they are all generated by the quarks which are likely just maintaining a power series. I remove pi from the ratios as that is likely the unknown.  Space impedance, for example, tells me that the density of charge will make a sphere is it exceeds a certain ratio, otherwise light leaks out along a straight, one degree of freedom line. Hence the Maxwell equations treated as a spectral problem with local wave action, no relativity, no time, no distance.