No loops required!
summin n = n * (n + 1) * (2*n + 1) `div` 6
summax n = n*n*n - summin (n - 1)
sumsum n = n*n*n + n*n
I'll put my thought process here because those equations look pretty arbitrary.
From that example grid of values of f, you can see the sum is 1*(n2 - (n-1)2) + 2 * ((n-1)2 - (n-2)2) + ... when you expand that out you get the sum of the first n squares. Which is the formula in summin (that Google lead me to).
For g, you can produce a chart of its value by taking an nxn square of n's, then subtracting the area of each square smaller than that. e.g. n*n*n - (n - 1)2 - (n - 2)2 -... which is the same as n*n*n - summin (n-1)
sumsum could be just the sum of the other two functions, but that's too simple. if you math it out, you get n*n*n + n*n. Much better.
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