Finding the total amount of blocks that needs to be moved

Let's say there are $n$ pistons. The $n^{th}$ piston will move $0$ blocks, the $(n-1)^{th}$ piston will move 1 block, the $(n-2)^{th}$ piston will move blocks. The block to be moved will be moved $n$ blocks. This means that the total number of blocks that needs to be displaced is : $$ToBeMoved(n)=n+(n-1)+(n-2)+ ... + 1$$ which is equivalent to $$ToBeMoved(n)=\frac{(n^{2} + n)}{2}$$

Minimizing the number of extensions

To minimize the number of moves, the pistons should push 12 blocks whenever they can. Imagine $27$ pistons. The piston that can push the most amount of blocks is the $12^{th}$ one. Then the $24^{th}$ one. We can push the $27^{th}$ piston. We now have a line of $26$ pistons and have made $3$ extensions so far. We can start building a function that takes as input the number of pistons and outputs the number of extensions needed. $$f(n)=\lceil \frac{n}{12} \rceil + f(n-1)$$ The number added after each call in the recursion $\lceil \frac{n}{12} \rceil$ if you think about it. At $12$ pistons and less you can just push the last piston to go to the next call of the recursion. At $13$ through $24$ you need to do $2$ extensions and so on

De-recursing the function

Calculating the recursion can be tidious by hand. To de-recurse it, think of the extensions that will be made.

Example for 27 (reads left to right then top to bottom):

12 24 27
12 24 26
12 24 25
12 24
12 23
12 22
12 21
12 20
12 19
12 18
12 17
12 16
12 15
12 14
12 13
12
11
10
9
8
7
6
5
4
3
2
1

You can arrange these in blocks. There's a block with $12\cdot1$ extensions, a block with $12 \cdot 2$ and a $3 \cdot 3$ block. This can be rewritten as $$f(27) = 3 \cdot 3 + 12(2 + 1)$$ More generally for $n$, this can be rewritten as $$f(n) = (n\ mod\ 12) \lceil \frac{n}{12} \rceil + 12 \cdot (\frac{(\lceil \frac{n}{12} \rceil - 1)\cdot \lceil\frac{n}{12} \rceil}{2})$$ $$f(n) = (n\ mod\ 12)\lceil \frac{n}{12} \rceil + 6\cdot(\lceil \frac{n}{12} \rceil^2-\lceil \frac{n}{12} \rceil)$$ $$f(n) = ((n\ mod\ 12) - 6)\lceil \frac{n}{12} \rceil + 6\lceil \frac{n}{12} \rceil^2$$ python3 code to generate the piston extension sequence would be

def piston_sequence(n):
	sequence = []
	for i in range(n, 0, -1):
		for j in range(12, i, 12):
			sequence.append(j)
		sequence.append(i)
	return sequence