Math: give you length 1~n line segments the longest edge length of the triangle does not exceed the number of your n triangles

Triangle Counting time limit: +Ms | Memory Limit:65535KB Difficulty:3

Describe

You are given n rods of length 1, 2 ..., N. You have the pick any 3 of them and build a triangle. How many distinct triangles can do? Note that, the triangles would be considered different if they has at least 1 pair of arms with different length.

Input

The input for each case would have

a single positive integer n (1<=n<=1000000). The end of input would be indicated by a case with n<1. This case is should not being processed.

Output

for the all test case, print the number of distinct triangles your can make.

Sample input

580

Sample output

322

Source

UVA

Uploaded by

Tc_ Li Yuanhai

Ideas:

The conventional idea is to simulate

But the complexity of the time will be (o^3) the obvious timeout

So we're guessing there must be a pattern, from a mathematical point of view.

Set the longest edge of the triangle to X, (other y,z) a maximum of C (x) triangles

So Y+z>x

X-y<z<x

So when Y=1 is misunderstood 0 y=2 a solution y=3 three solutions .....

0+1+2+3+....+ (x-2)

Total: (x-2) * (x-1)/2

But this is where the y=z is calculated and two times per triangle.

(Y=z situation) y=x/2+1 start (x-1)-(x/2+1) +1=x/2-1 not difficult to find X for odd case is X/2 even case is x/2-1 so→ (tip) (X-1)/2 Avoid parity discussion

So C (x) = ((x-1) (x-2)/2-(x-1)/2)/2

The problem is the number of so F (n) =f (n-1) +c (n) of triangles with the longest side length not exceeding n;

#include <iostream> #include <stdio.h> #include <string.h> #include <algorithm>using namespace Std;long long A[1000005];int Main () {    int n;    a[1]=a[2]=a[3]=0;    For (long long i=4;i<1000005;i++)        a[i]=a[i-1]+ ((i-1) * (i-2)/2-(i-1)/2)/2;    while (~SCANF ("%d", &n)    } {        if (n<=0) break            ;        printf ("%lld\n", A[n]);}    }

Math: give you length 1~n line segments the longest edge length of the triangle does not exceed the number of your n triangles