CF914G Sum the Fibonacci（FWT模板+子集卷积）

title

题目

solution

$(s_a|s_b)\&s_c\&(s_d\oplus s_e)=2^i,i\in Z;s_a\&s_b=0$
令$i=s_a|s_b,j=s_c,d=s_d\oplus s_e$

$$\sum _x\sum _{i\&j\&k=x}fi{b}_i\ast fi{b}_j\ast fi{b}_k\ast (\sum _{s_a|s_b=i,s_a\&s_b=0}1)\ast (\sum _{s_c=j}1)\ast (\sum _{s_d\oplus s_e=k}1)$$

令$f_i=({\sum }_{s_a|s_b=i,s_a\&s_b=0}1),g_j=({\sum }_{s_c=j}1),h_k=({\sum }_{s_d\oplus s_e=k}1)$
$f$就是子集卷积直接做；$g$不做操作；$h$就是常规的$FWT\_xor$卷积
最后三个一起带着各自的斐波拉契权值，做个$FWT\_and$卷积即可

code

#include <cstdio>
#include <cstring>
#define mod 1000000007
#define int long long
#define maxn 1 << 18
int n, ret, inv = ( mod + 1 ) >> 1;
int f[maxn], g[maxn], h[maxn], tmp[maxn], ans[maxn], fib[maxn];
int dp[20][maxn];

void FWT_or( int *v, int opt ) {
    
    
	for( int i = 1;i < maxn;i <<= 1 )
		for( int j = 0;j < maxn;j += ( i << 1 ) )
			for( int k = 0;k < i;k ++ )
				v[j + k + i] = ( v[j + k + i] + v[j + k] * opt + mod ) % mod;
}

void FWT_and( int *v, int opt ) {
    
    
	for( int i = 1;i < maxn;i <<= 1 )	
		for( int j = 0;j < maxn;j += ( i << 1 ) )
			for( int k = 0;k < i;k ++ )
				v[j + k] = ( v[j + k] + v[j + k + i] * opt + mod ) % mod;;
}

void FWT_xor( int *v, int opt ) {
    
    
	for( int i = 1;i < maxn;i <<= 1 )
		for( int j = 0;j < maxn;j += ( i << 1 ) )
			for( int k = 0;k < i;k ++ ) {
    
    
				int x = v[j + k], y = v[j + k + i];
				v[j + k] = ( x + y ) % mod;
				v[j + k + i] = ( x - y + mod ) % mod;
				if( opt == -1 ) {
    
    
					v[j + k] = v[j + k] * inv % mod;
					v[j + k + i] = v[j + k + i] * inv % mod;
				}
			}
}

signed main() {
    
    
	scanf( "%lld", &n );
	for( int i = 1, x;i <= n;i ++ ) {
    
    
		scanf( "%lld", &x );
		f[x] ++, g[x] ++, h[x] ++;
	}
	fib[1] = 1;
	for( int i = 2;i < maxn;i ++ )
		fib[i] = ( fib[i - 1] + fib[i - 2] ) % mod;
	for( int i = 0;i < maxn;i ++ )
		dp[__builtin_popcount( i )][i] = f[i], f[i] = 0;
	n = 18;
	for( int i = 0;i < n;i ++ )
		FWT_or( dp[i], 1 );
	for( int i = 0;i < n;i ++ ) {
    
    
		memset( tmp, 0, sizeof( tmp ) );
		for( int j = 0;j <= i;j ++ )
			for( int k = 0;k < maxn;k ++ )
				tmp[k] = ( tmp[k] + dp[j][k] * dp[i - j][k] % mod ) % mod;
		FWT_or( tmp, -1 );
		for( int j = 0;j < maxn;j ++ )
			if( __builtin_popcount( j ) == i )
				f[j] = ( f[j] + tmp[j] ) % mod;
	}
	FWT_xor( h, 1 );
	for( int i = 0;i < maxn;i ++ )
		h[i] = h[i] * h[i] % mod;
	FWT_xor( h, -1 ); 
	for( int i = 0;i < maxn;i ++ )
		f[i] = f[i] * fib[i] % mod, g[i] = g[i] * fib[i] % mod, h[i] = h[i] * fib[i] % mod;
	FWT_and( f, 1 ), FWT_and( g, 1 ), FWT_and( h, 1 );
	for( int i = 0;i < maxn;i ++ )
		ans[i] = f[i] * g[i] % mod * h[i] % mod;
	FWT_and( ans, -1 );
	for( int i = 1;i < maxn;i <<= 1 )
		ret = ( ret + ans[i] ) % mod;
	printf( "%lld\n", ret );
	return 0;
}