A pattern of signs

Take any four consecutive integers $a, b, c, d$. Then it is obvious that $a+d = b+c$. Let us write this relation as 
\[+a-b-c+d = 0.\]
Note the sequence of signs on the left side: $+,-,-,+$. We shall write the above statement in the following form: 

The sign sequence $+--+$ if applied to any $4$ consecutive integers yields a sum of $0$.

For example, if the 4 consecutive integers are $12$, $13$, $14$, $15$ we have: 
\[+ 12 - 13 - 14 + 15 = 0.\]
Now reverse all the signs in this string; we get the string $-++-$. If we concatenate the two strings together we get the string $+--+-++-$ (which is now of length $8$). Now here is a surprising fact:

The sign sequence $+--+-++-$ if applied any $8$ consecutive squares yields a sum of $0$.

For example, take the $8$ consecutive squares $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$. We have now:
\[+ 9 - 16 - 25 + 36 - 49 + 64 + 81 - 100 = 0.\]
Please verify this statement. Then try it out on other collections of $8$ consecutive squares.


Now take the string $+--+-++-$ and reverse all its signs; we get the string $-++-+--+$. If we concatenate these two strings together, we get the string $+--+-++--++-+--+$ (which is now of length $16$). Here is the next surprising fact:

The sign sequence $+--+-++--++-+--+$ if applied any $16$ consecutive cubes yields a sum of $0$.

Here is an example: take the $16$ consecutive cubes $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$, $1331$, $1728$, $2197$, $2744$, $3375$, $4096$, $4913$, $5832$. Applying the sign sequence just given to these numbers, you will find you get a sum of $0$. I have not written the full "sum" here as it does not fit into this space properly: $+ 27 - 64 - 125 + 216 - 343 + 512 + 729 - 1000 - 1331 + 1728 + 2197 -2744 + 3375 - 4096 - 4913 + 5832 = 0$. But please check it out.

Then try this out for other collections of $16$ consecutive cubes.


I suppose you will now be able to guess what comes next in this sequence of statements .... 


But why do we have such a pattern?