Triple Sums

You‘re given a sequence s of N distinct integers.
Consider all the possible sums of three integers from the sequence at three different indicies.
For each obtainable sum output the number of different triples of indicies that generate it.

Constraints:
N <= 40000, |s_i| <= 20000

Input

The first line of input contains a single integer N.

Output

Print the solution for each possible sum in the following format:
sum_value : number_of_triples

Smaller sum values should be printed first.

Example

Input:

5
-1
2
3
0
5
Output:

1 : 1
2 : 1
4 : 2
5 : 1
6 : 1
7 : 2
8 : 1
10 : 1

Explanation:
4 can be obtained using triples ( 0, 1, 2 ) and ( 0, 3, 4 ).

題意：

　　給你n個數，問你任選三個不同序號的數和為x的方案數有多少

題解：

　　FFT；

　　容斥原理要學好

#include<bits/stdc++.h>
using namespace std;
#pragma comment(linker, "/STACK:102400000,102400000")
#define ls i<<1
#define
 
 rs ls | 1
#define mid ((ll+rr)>>1)
#define pii pair<int,int>
#define MP make_pair
typedef long long LL;
const long long INF = 1e18+1LL;
const double pi = acos(-1.0);
const int N = 5e5+10, M = 1e3+20,inf = 2e9,mod = 1e9+7;


struct Complex {
    double r , i ;
    Complex () {}
    Complex ( double r , double i ) : r ( r ) , i ( i ) {}
    Complex operator + ( const Complex& t ) const {
        return Complex ( r + t.r , i + t.i ) ;
    }
    Complex operator - ( const Complex& t ) const {
        return Complex ( r - t.r , i - t.i ) ;
    }
    Complex operator * ( const Complex& t ) const {
        return Complex ( r * t.r - i * t.i , r * t.i + i * t.r ) ;
    }
} ;

void FFT ( Complex y[] , int n , int rev ) {
    for ( int i = 1 , j , t , k ; i < n ; ++ i ) {
        for ( j = 0 , t = i , k = n >> 1 ; k ; k >>= 1 , t >>= 1 ) j = j << 1 | t & 1 ;
        if ( i < j ) swap ( y[i] , y[j] ) ;
    }
    for ( int s = 2 , ds = 1 ; s <= n ; ds = s , s <<= 1 ) {
        Complex wn = Complex ( cos ( rev * 2 * pi / s ) , sin ( rev * 2 * pi / s ) ) , w ( 1 , 0 ) , t ;
        for ( int k = 0 ; k < ds ; ++ k , w = w * wn ) {
            for ( int i = k ; i < n ; i += s ) {
                y[i + ds] = y[i] - ( t = w * y[i + ds] ) ;
                y[i] = y[i] + t ;
            }
        }
    }
    if ( rev == -1 ) for ( int i = 0 ; i < n ; ++ i ) y[i].r /= n ;
}

int num[N],n,x,now[N];
Complex s[N*8],t[N*8],tt[N];
int main() {
    scanf("%d",&n);
    for(int i = 1; i <= n; ++i) {
        scanf("%d",&x);
        num[x + 20000]++;
    }
    int n1;
    for(int i = 1; i <= 20000*6; i<<=1,n1=i);

    for(int i = 0; i <= 20000*2; ++i) now[i+i] += num[i];
    for(int i = 0; i <= 20000*4; ++i) s[i] = Complex(now[i],0);
    for(int i = 20000*4+1; i < n1; ++i) s[i] = Complex(0,0);

    for(int i = 0; i <= 2*20000; ++i) t[i] = Complex(num[i],0);
    for(int i = 2*20000+1; i < n1; ++i) t[i] = Complex(0,0);
    for(int i = 0; i < n1; ++i) tt[i] = t[i];
    FFT(s,n1,1),FFT(t,n1,1);FFT(tt,n1,1);
    for(int i = 0; i < n1; ++i) t[i] = t[i]*t[i]*t[i];
    for(int i = 0; i < n1; ++i) s[i] = s[i]*tt[i];
    FFT(s,n1,-1),FFT(t,n1,-1);
    int cnt = 0;
    for(int i = 0; i <= 6*20000; ++i) {
        int x = ((int)(t[i].r + 0.5)) - 3*((int)(s[i].r+0.5));
        if(i%3==0) x += 2*num[i/3];
        x/=6;
        if(x) {
            printf("%d : %d\n",i - 3*20000,x);
        }
    }
    return 0;
}

SPOJ - TSUM Triple Sums FFT+容斥