Generating noble gases

The previous post discussed what the periodic table would look like if it could be extended indefinitely and if certain patterns in the actual table continued to hold. In particular, the last element of each period would have atomic number

Z_n = \frac{(-1)^n(3n+6) + 2n^3 + 12n^2 + 25n - 6}{12}

and so we could call the Zn in the equation above noble numbers, atomic numbers of noble gases.

We could then look at the generating function for the noble numbers by multiplying each Zn by xn and summing.

g(x) = \sum_{n=1}^\infty Z_nx^n

There’s no practical reason I can imagine for computing this other than it provides an excuse to play around with generating functions.

Does the series above converge? For some purposes it doesn’t matter. Generating functions are often useful as formal sums even if if they do not converge as analytic functions.

However, this function does converge. The Zn grow like a polynomial function of n, and for |x| < 1, the xn terms decrease exponentially as a function of n, so the series converges. The series has radius of convergence 1 because it clearly diverges for |x| > 1.

We can do better than saying the series converges: we can find the sum in closed form. We’ll use Mathematica to do the calculation for us.

    Sum[((-1)^n (3 n + 6) + 2 n^3 + 12 n^2 + 25 n - 6 ) x^n, 
        {n, 1, Infinity}]/12

This returns the following rational function:

\frac{2 x \left(x^4-x^3-2 x^2+3 x+1\right)}{(x-1)^4 (x+1)^2}

Note that the generating function has singularities at -1 and 1. This confirms our argument above that the radius of convergence should be 1.

We can use Mathematica to expand the generating function as a series, and we should recover the atomic numbers of the noble gases.

    Series[g[x], {x, 0, 7}]

And indeed we do:

2 x+10 x^2+18 x^3+36 x^4+54 x^5+86 x^6+118 x^7+O\left(x^8\right)

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