In a standard square grid pattern the distances to integer root locations is simply the sum of two squares. We find that these distances have $\sqrt1, \sqrt2, \sqrt4, \sqrt5, \sqrt8, \sqrt9, \sqrt10, \sqrt13, ...$ But, knowing it's the sum of two squares we know this is Fermat's theorem on the point and that these exist so long as the prime factors of the number are even for $4k + 3$. So $3$ is absent because $3$ has prime factors of $3$ and there's 1 $3$ which is odd and $4*0+3 = 3$. $6$ is missing as it has prime factors $2,3$, which is again an odd number of $3$s. $7$ is also missing as $4*1+3 = 7$ and the prime factors of $7$ are $7, 1$. We also don't have $11$, as $11$ is $4*2+3 = 11$. $12$ is gone because prime factors are $3,2,2$ which is an odd number of 3s.
In this case the root-integer distances are the same as the sum of squares, and so we see the pattern.
Now, to the question: is there a similar pattern for an isometric grid? And if so what? I mean it's missing $\sqrt2, \sqrt5, \sqrt6, \sqrt8, \sqrt10, \sqrt11$ but why?
For further info on the background of the question: http://www.fractalcurves.com/Root6.html