how to count frequency count in triple nested loop

Question

I am trying to determine the frequency count of a triple nested loop.

for i = 1 to n do
    for j = 1 to i do
        for k = i to j do
            x =  x + 1

I know that the statement x = x + 1 will not get executed until i attains the value of n

Any tips/suggestions on how to get started?

Answer 1

Let's take 4 and 5 as examples. When i = 4,

...
    for j = 1 to 4 do
        for k = 4 to j do
            x =  x + 1

...j = 1
   for k = 4 to 1 do  // 4 times
       x =  x + 1
...j = 2
   for k = 4 to 2 do  // 3 times
       x =  x + 1
...j = 3
   for k = 4 to 3 do   // twice
       x =  x + 1
...j = 4
   for k = 4 to 4 do   // once
       x =  x + 1

When i = 5,

...
    for j = 1 to 5 do
        for k = 5 to j do
            x =  x + 1

...j = 1
   for k = 5 to 1 do  // 5 times
       x =  x + 1
...j = 2
   for k = 5 to 2 do  // 4 times
       x =  x + 1
...j = 3
   for k = 5 to 3 do   // 3 times
       x =  x + 1
...j = 4
   for k = 5 to 4 do   // twice
       x =  x + 1
...j = 5
   for k = 5 to 5 do   // once
       x =  x + 1

pattern?