This question originates the following observations about the fine structure constant, $alpha$.

Measurements of $alpha^{-1}$ yield values slightly smaller than $137.036$, and a good rational fit for all of them is $frac{34259}{250}$. The choice of $250$ does not seem to be decimally biased, because one can multiply any value close to $137.035999$ for all integers up to say $300$ and $250$ emerges as a proper denominator more clearly than $113$ for $pi$.

One of the alternative approximate expressions for $alpha^{-1}$ is $$5^3+frac{5^2}{2}-frac{1}{2}+5^{-2}-frac{5^{-3}}{2}$$ After some easy manipulation, this can be written as $$frac{1}{2^3}(10^3+2^3ยท3(1+frac{3}{10^3}))$$

I would like to learn how to decide between structure or coincidence, based on data. For instance, how should one compare the complexities of the approximate value of the constant ($137.036$) and the above expression that suggests some internal structure?

An equivalent way to see $137$ (only the integer part of $alpha^{-1}$) is the following image, corresponding to the polynomial $(4n+1)^3+3(2n)^2$ at $n=1$.

I would classify this as a figurate number, as it is the sum of related (by $n$) figurate numbers (even squares and related cubes). But how can the *structureness* of this representation be evaluated?