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 $$\frac{1}{2^3}(10^3+2^3·3(1+\frac{3}{10^3}))$$
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 $n=1$, or $(2n+1)^3+3n^2$ at $n=2$.
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?
Working with spheres instead of cubes, the correction $.036$ may be obtained from the normalized volume of a sphere of radius $5$ with $24·5^6$ spheres of radius $\frac{1}{25}$ centered on its surface, according to the following volume integral
$$\frac{4}{3}\alpha^{-1} \approx \frac{4}{3}\times137.036 = \int_{-10}^{0} (5^2 - (x + 5)^2) dx $$
$$+ 24·5^6 \left( \int_{-\frac{1}{10·25^2}}^{\frac{1}{25}} \left(\frac{1}{25^2} - x^2\right) dx - \int_{-\frac{1}{10·25^2}}^{0} \left(5^2 - (x + 5)^2\right) dx\right)$$