Counting arrays with Euclidean distance at most 2 from a given binary array I have a binary array like this: 

$$A = [0,1,0,0,1,0]\,.$$

I'm trying to find a way to calculate how many arrays of the same length exist that have a Euclidean distance of 2 or less from this array.

So, how many arrays of length 6 exist where

$$\sqrt{\Sigma(A_{_i} - B_{_i})^2}\leq 2\,?$$

I'm trying to find or create a formula that takes an array like above and outputs a count of how many possible binary arrays exist that fit the conditions of the above formula.

I've looked online for a formula without success.
Don't search for a formula&nbsp;&ndash; you'll probably never find something so specific. Instead, try to break up the task into smaller units.

Since your arrays are binary,
$$(A_i-B_i)^2 = \begin{cases}0 &amp;\text{if }A_i=B_i\\ 1&amp;\text{if }A_i\neq B_i\,.\end{cases}$$

So $\sqrt{\sum(A_i-B_i)^2}\leq 2$ if, and only if there are at most four values of&nbsp;$i$ such that $A_i\neq B_i$.  So you just need to compute the number of ways that $B$&nbsp;could have zero, one, two, three or four different entries from&nbsp;$A$ (or, more simply, the number of ways that it could have five or six different entries, and subtract that from the total possible values of&nbsp;$B$). This calculation is high-school combinatorics. Note that, because the arrays are binary, there's only one value of&nbsp;$B_i$ that's the same as&nbsp;$A_i$ and only one that's different.