Handling the Bohr model

I am trying to match counting directions for quants with the Bohr model. This is about aligning the axis between this and the Lucas model and converting the total energy of the Bohr model into delta energy per level in the Lucas model. On top of this is the angle matching between hyperbolic and trigonometric with the Lucas model. Bohr also counts negative energy, the amount of energy needed to remove the electron.   I am trying to straighten this out, see below.

The energy of the n-th level for any atom is determined by the radius and quantum number/ An electron in the lowest energy level of hydrogen (n = 1) therefore has about 13.6 eV less energy than a motionless electron infinitely far from the nucleus. The next energy level (n = 2) is −3.4 eV. The third (n = 3) is −1.51 eV, and so on. For larger values of n, these are also the binding energies of a highly excited atom with one electron in a large circular orbit around the rest of the atom.

Lets try to nail down down direction in counting Lucas numbers.

1) The highest order Lucas number matches the Fine Structure.
2) The fine structure constant must be the high end band limit of light from the electron orbital. Isn't that limit a radial frequency?

3) The atom gets heavy toward the center, high Compton frequency goes with higher heaviness,  λ=h/mc and high Gamma frequency comes from the atam center.

4) The Lucas quants counts layers, it is not an integral.  I mean light is a power spectra and all of its cylinders are firing. So the Lucas atom will be a power series, going from the exterior to the center.

That means Lucas numbers count up from the outer shell toward the more inert center. This is the way I have been counting exponents, but Hyperbolic says we can count symmetrical anyway. 

And, the Lucas polynomials are equally defined over the interval zero to one, and are not a radial function, so that variable has to be scaled.  And the hyperbolic angle will be scaled.

I likely got a lot of this backward, but that is going to be my first effort, so you experts out there, feel free to jump in.