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I was once at a party, and i was somewhat bored and i found this way of calculating the next square. It works without multiplication, so its suitable for mental calculation.

Seeing that i have recently learned python, here’s a python version of it:

n = 10 # how many sqs to return

b = []
def sq(x):
    return x*x
for y in range(1,n):
    print sq(y)
    b.append(sq(y))


def sqx(x):
    if x == 1:
        return 1
    if x == 2:
        return 4
    return (sqx(x-1)-sqx(x-2))+sqx(x-1)+2

a = []
for y in range (1,n):
    print sqx(y)
    a.append(sqx(y))

In english. First, set the first two squares to 1 and 4, since this method needs to use the two previous squares to calculate the next. Then calculate the absolute difference between these two. Suppose we are looking for 3², so previous two are 1 and 4. Abs diff is 3. Add 2 to this, result 5. Add 5 to previous square, so 4+5=9. 9 is 3².

I have no idea why this works, i just saw a pattern, and confirmed it for the first 20 integers or so.

In the code above, i have defined the function recursively. It is much slower than the other function. I suppose both are slower than the low-level premade function pow(n,m). But it certainly is cool. :P