Consider integer coordinates x, y in the Cartesian plan and three functions f, g, h defined by:
f: 1 <= x <= n, 1 <= y <= n --> f(x, y) = min(x, y)
g: 1 <= x <= n, 1 <= y <= n --> g(x, y) = max(x, y)
h: 1 <= x <= n, 1 <= y <= n --> h(x, y) = x + y
where n is a given integer (n >= 1, guaranteed) and x, y are integers.
In the table below you can see the value of the function f with n = 6.
| --- | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| 6 | - | 1 | 2 | 3 | 4 | 5 | 6 |
| 5 | - | 1 | 2 | 3 | 4 | 5 | 5 |
| 4 | - | 1 | 2 | 3 | 4 | 4 | 4 |
| 3 | - | 1 | 2 | 3 | 3 | 3 | 3 |
| 2 | - | 1 | 2 | 2 | 2 | 2 | 2 |
| 1 | - | 1 | 1 | 1 | 1 | 1 | 1 |
| 0 | - | - | - | - | - | - | - |
The task is to calculate the sum of f(x, y), g(x, y) and h(x, y) for all integers x and y such that (1 <= x <= n, 1 <= y <= n).
The function sumin (sum of f) will take n as a parameter and return the sum of min(x, y) in the domain 1 <= x <= n, 1 <= y <= n. The function sumax (sum of g) will take n as a parameter and return the sum of max(x, y) in the same domain. The function sumsum (sum of h) will take n as a parameter and return the sum of x + y in the same domain.
sumin(6) --> 91
sumin(45) --> 31395
sumin(999) --> 332833500
sumin(5000) --> 41679167500
sumax(6) --> 161
sumax(45) --> 61755
sumax(999) --> 665167500
sumax(5000) --> 83345832500
sumsum(6) --> 252
sumsum(45) --> 93150
sumsum(999) --> 998001000
sumsum(5000) --> 125025000000
- Try to avoid nested loops
- Note that h = f + g