Generalizing hyperbolics

I am working the general solution to:
cosh^2-sinh^2 = Y
Y generally greater than one. But I work it from the cosh function so the solution is alwasy symmetric around Y or 1/Y.

So divide through by Y and change the base from Euler to 1/sqrt(Y), then quantize the exponent in integers.  The greater Y is from one, the fewer quants until the tanh curve is full. It works like Fermi Dirac:

For the distribution of coth'', and works like the bosons for tanh''.  So making Y greater than one reduces the supported exchange rate, kT,  as in the plot.  That is lower energy, all the fermions crowd near the low hyperbolic angles.  Setting Y to Phi generates the Lucas sequence, and I suspect I can generate the silver ratio sequence, or any of the Lagrange sequences.