多米诺骨牌(domino)
https://dsa.cs.tsinghua.edu.cn/oj/problem.shtml?id=186
问题描述
小牛牛对多米诺骨牌有很大兴趣，然而她的骨牌比较特别，只有黑色和白色的两种。她觉得如果存在连续三个骨牌是同一种颜色，那么这个骨牌排列便是不美观的。现在她有n个骨牌要来排列，她想知道不美观的排列的个数。由于数字较大，数学不好的她不会统计，所以请你来帮忙。希望你帮她求出不美观的排列的个数。
输入数据
只有一个正整数，即要排列的骨牌个数。
输出数据
一个数，即不美观的排列个数。
样例输入
4
样例输出
6
样例解释
有 6 种不美观的排列。
黑黑黑黑，白白白白，黑黑黑白，白白白黑，黑白白白，白黑黑黑
数据范围
20%的数据，n<=60；
50%的数据，n<=6000；
100%的数据，n<=10000。
时间限制： 1 sec
空间限制： 256 MB
提示
动态规划、高精度加法。
注意：开头部分定义的
#define MAXN 9999
#define MAXSIZE 500 //最大长度
#define DLEN 4
class BigNum
{
private:
int a[1000]; //控制大数的位数
int len; //长度
}
对算法的正确性影响很大
#include<iostream>
#include<cstdio>
#include<string.h>
#include<cmath>
#include<sstream>
using namespace std;
#define MAXN 9999
#define MAXSIZE 500 //最大长度
#define DLEN 4
class BigNum
{
private:
int a[1000]; //控制大数的位数
int len; //长度
public:
BigNum(){ len = 1;memset(a,0,sizeof(a)); } //构造函数
void XD();
BigNum(const int);
BigNum(const long long int);
BigNum(const char*);
BigNum(const string &);
BigNum(const BigNum &); //拷贝构造函数
BigNum &operator = (const BigNum &); //重载赋值运算符
BigNum &operator = (const int &);
BigNum &operator = (const long long int &);
friend istream& operator >> (istream&, BigNum&); //重载输入运算符
friend ostream& operator << (ostream&, BigNum&); //重载输出运算符
template<typename T> BigNum operator << (const T &) const;
template<typename T> BigNum operator >> (const T &) const;
BigNum operator + (const BigNum &) const; //重载加法运算符，大数加大数
BigNum operator - (const BigNum &) const; //重载减法运算符，大数减大数
BigNum operator * (const BigNum &) const; //重载乘法运算符，大数乘大数
bool operator > (const BigNum& b)const; //重载大于
bool operator < (const BigNum& b) const; //重载小于
bool operator == (const BigNum& b) const; //重载等于符号
template<typename T> BigNum operator / (const T &) const; //重载除法运算符，大数除整数
template<typename T> BigNum operator ^ (const T &) const; //大数的n次方
template<typename T> T operator % (const T &) const; //大数对int取模
template<typename T> BigNum operator + (const T& b) const {BigNum t = b; t = *this + t; return t;}
template<typename T> BigNum operator - (const T& b) const {BigNum t = b; t = *this - t; return t;}
template<typename T> BigNum operator * (const T& b) const {BigNum t = b; t = (*this) * t; return t;}
template<typename T> bool operator < (const T& b) const {BigNum t = b; return ((*this) < t);}
template<typename T> bool operator > (const T& b) const {BigNum t = b; return ((*this) > t);}
template<typename T> bool operator == (const T& b) const {BigNum t = b; return ((*this) == t);}
bool operator <= (const BigNum& b) const {return (*this) < b || (*this) == b;}
bool operator >= (const BigNum& b) const {return (*this) > b || (*this) == b;}
bool operator != (const BigNum& b) const {return !((*this) == b);}
template<typename T> bool operator >= (const T& b) const {BigNum t = b; return !((*this) < t);}
template<typename T> bool operator <= (const T& b) const {BigNum t = b; return !((*this) > t);}
template<typename T> bool operator != (const T& b) const {BigNum t = b; return !((*this) == t);}
BigNum& operator += (const BigNum& b) {*this = *this + b; return *this;}
BigNum& operator -= (const BigNum& b) {*this = *this - b; return *this;}
BigNum& operator *= (const BigNum& b) {*this = *this * b; return *this;}
template<typename T> BigNum& operator /= (const T& b) {*this = *this/b; return *this;}
template<typename T> BigNum& operator %= (const T& b) {*this = *this%b; return *this;}
template<typename T> BigNum& operator += (const T& b) {*this = *this+b; return *this;}
template<typename T> BigNum& operator -= (const T& b) {*this = *this-b; return *this;}
template<typename T> BigNum& operator *= (const T& b) {*this = *this*b; return *this;}
template<typename T> BigNum& operator ^= (const T& b) {*this = *this^b; return *this;}
BigNum operator ++ (int) {BigNum t = *this; *this += 1; return t;}
BigNum operator -- (int) {BigNum t = *this; *this -= 1; return t;}
BigNum& operator -- () {*this -= 1; return *this;}
BigNum& operator ++ () {*this += 1; return *this;}
template<typename T> BigNum& operator <<= (const T& b) {*this = *this << b; return *this;}
template<typename T> BigNum& operator >>= (const T& b) {*this = *this >> b; return *this;}
template<typename T> BigNum friend operator + (const T& a, const BigNum& b) {BigNum t = a; t = t + a; return t;}
template<typename T> BigNum friend operator - (const T& a, const BigNum& b) {BigNum t = a; t = t - b; return t;}
template<typename T> BigNum friend operator * (const T& a, const BigNum& b) {BigNum t = a; t = t * b; return t;}
template<typename T> friend bool operator < (const T& a, const BigNum& b) {return b > a;}
template<typename T> friend bool operator > (const T& a, const BigNum& b) {return b < a;}
template<typename T> friend bool operator <= (const T& a, const BigNum& b) {return b >= a;}
template<typename T> friend bool operator >= (const T& a, const BigNum& b) {return b <= a;}
template<typename T> friend bool operator == (const T& a, const BigNum& b) {return b == a;}
template<typename T> friend bool operator != (const T& a, const BigNum& b) {return b != a;}
void print(); //输出大数
int Size(); //返回大数长度
int the_first(); //返回第一个数字
int the_last(); //返回最后一位数字
int to_int(); //转化为整数
long long int to_long();
string to_String(); //转化为string类型
//char* to_char();
};
BigNum::BigNum(const int b) //将一个int类型的变量转化为大数
{
int c,d = b;
len = 0;
memset(a,0,sizeof(a));
while(d > MAXN){
c = d - (d / (MAXN+1)) * (MAXN+1);
d = d / (MAXN+1);
a[len++] = c;
}
a[len++] = d;
}
BigNum::BigNum(const long long int b)
{
long long int c,d = b;
len = 0;
memset(a,0,sizeof(a));
while(d > MAXN){
c = d - (d / (MAXN+1)) * (MAXN+1);
d = d / (MAXN+1);
a[len++] = c;
}
a[len++] = d;
}
BigNum::BigNum(const string& s)
{
int t,k,index,l,i;
memset(a,0,sizeof(a));
l = s.size();
len = l/DLEN;
if(l%DLEN)
len++;
index = 0;
for(i = l-1; i >=0 ;i -= DLEN){
t = 0;
k = i-DLEN+1;
if(k < 0) k = 0;
for(int j = k; j <= i; j++)
t = t*10 + s[j]-'0';
a[index++] = t;
}
}
BigNum::BigNum(const char* s) //将一个字符串类型的变量转化为大数
{
int t,k,index,l,i;
memset(a,0,sizeof(a));
l = strlen(s);
len = l/DLEN;
if(l%DLEN)
len++;
index = 0;
for(i = l-1; i >= 0; i -= DLEN){
t = 0;
k = i - DLEN + 1;
if(k < 0) k = 0;
for(int j = k; j <= i; j++)
t = t*10 + s[j] - '0';
a[index++] = t;
}
}
BigNum::BigNum(const BigNum & b) : len(b.len) //拷贝构造函数
{
memset(a,0,sizeof(a));
for(int i = 0 ; i < len ; i++)
a[i] = b.a[i];
}
BigNum & BigNum::operator = (const BigNum& n) //重载赋值运算符，大数之间进行赋值运算
{
len = n.len;
memset(a,0,sizeof(a));
for(int i = 0 ; i < len ; i++)
a[i] = n.a[i];
return *this;
}
BigNum & BigNum::operator = (const int& num)
{
BigNum t(num);
*this = t;
return *this;
}
BigNum & BigNum::operator = (const long long int& num)
{
BigNum t(num);
*this = t;
return *this;
}
void XD()
{
cout << "A hidden egg! Good luck for u!" << endl;
}
istream& operator >> (istream & in, BigNum & b) //重载输入运算符
{
char ch[MAXSIZE*4];
int i = -1;
in>>ch;
int l = strlen(ch);
int cnt = 0, sum = 0;
for(i = l-1; i >= 0; ){
sum = 0;
int t = 1;
for(int j = 0; j < 4 && i >= 0; j++,i--,t *= 10)
sum += (ch[i]-'0')*t;
b.a[cnt] = sum;
cnt++;
}
b.len = cnt++;
return in;
}
ostream& operator << (ostream& out, BigNum& b) //重载输出运算符
{
int i;
cout << b.a[b.len - 1];
for(i = b.len - 2 ; i >= 0 ; i--){
cout.width(DLEN);
cout.fill('0');
cout << b.a[i];
}
return out;
}
template<typename T> BigNum BigNum::operator << (const T& b) const
{
T temp = 1;
for(int i = 0; i < b; i++)
temp *= 2;
BigNum t = (*this) * temp;
return t;
}
template<typename T> BigNum BigNum::operator >> (const T& b) const
{
T temp = 1;
for(int i = 0; i < b; i++)
temp *= 2;
BigNum t = (*this) / temp;
return t;
}
BigNum BigNum::operator + (const BigNum& b) const //两个大数之间的相加运算
{
BigNum t(*this);
int i,big;
big = b.len > len ? b.len : len;
for(i = 0 ; i < big ; i++){
t.a[i] += b.a[i];
if(t.a[i] > MAXN){
t.a[i + 1]++;
t.a[i] -=MAXN+1;
}
}
if(t.a[big] != 0)
t.len = big + 1;
else
t.len = big;
return t;
}
BigNum BigNum::operator - (const BigNum& b) const //两个大数之间的相减运算
{
int i,j,big;
bool flag;
BigNum t1,t2;
if(*this>b){
t1 = *this;
t2 = b;
flag = 0;
}
else{
t1 = b;
t2 = *this;
flag = 1;
}
big = t1.len;
for(i = 0 ; i < big ; i++){
if(t1.a[i] < t2.a[i]){
j = i + 1;
while(t1.a[j] == 0)
j++;
t1.a[j--]--;
while(j > i)
t1.a[j--] += MAXN;
t1.a[i] += MAXN + 1 - t2.a[i];
}
else
t1.a[i] -= t2.a[i];
}
t1.len = big;
while(t1.a[t1.len - 1] == 0 && t1.len > 1){
t1.len--;
big--;
}
if(flag)
t1.a[big-1] = 0-t1.a[big-1];
return t1;
}
BigNum BigNum::operator * (const BigNum& b) const //两个大数之间的相乘运算
{
BigNum ret;
int i,j,up;
int temp,temp1;
for(i = 0 ; i < len ; i++){
up = 0;
for(j = 0 ; j < b.len ; j++){
temp = a[i] * b.a[j] + ret.a[i + j] + up;
if(temp > MAXN){
temp1 = temp - temp / (MAXN + 1) * (MAXN + 1);
up = temp / (MAXN + 1);
ret.a[i + j] = temp1;
}
else{
up = 0;
ret.a[i + j] = temp;
}
}
if(up != 0) ret.a[i + j] = up;
}
ret.len = i + j;
while(ret.a[ret.len - 1] == 0 && ret.len > 1)
ret.len--;
return ret;
}
template<typename T> BigNum BigNum::operator / (const T& b) const
{
BigNum ret;
T i,down = 0;
for(i = len - 1 ; i >= 0 ; i--){
ret.a[i] = (a[i] + down * (MAXN + 1)) / b;
down = a[i] + down * (MAXN + 1) - ret.a[i] * b;
}
ret.len = len;
while(ret.a[ret.len - 1] == 0 && ret.len > 1)
ret.len--;
return ret;
}
template<typename T> T BigNum::operator % (const T& b) const
{
T i,d=0;
for (i = len-1; i>=0; i--){
d = ((d * (MAXN+1))% b + a[i])% b;
}
return d;
}
template<typename T> BigNum BigNum::operator^(const T& n) const //大数的n次方运算
{
BigNum t,ret(1);
int i;
if(n < 0) return 0;
if(n == 0)
return 1;
if(n == 1)
return *this;
int m = n;
while(m > 1){
t =* this;
for(i = 1; (i<<1) <= m;i <<= 1)
t = t*t;
m-=i;
ret=ret*t;
if(m == 1) ret = ret * (*this);
}
return ret;
}
bool BigNum::operator > (const BigNum& b) const //大数和另一个大数的大小比较
{
int tot;
if(len > b.len)
return true;
else if(len == b.len){
tot = len - 1;
while(a[tot] == b.a[tot] && tot >= 0)
tot--;
if(tot >= 0 && a[tot] > b.a[tot])
return true;
else
return false;
}
else
return false;
}
bool BigNum::operator < (const BigNum& b) const
{
int tot;
if(len > b.len)
return false;
else if(len == b.len){
tot = len - 1;
while(a[tot] == b.a[tot] && tot >= 0)
tot--;
if(tot >= 0 && a[tot] > b.a[tot])
return false;
else
return false;
}
else
return true;
}
bool BigNum::operator == (const BigNum& b) const
{
int tot = len-1;
if(len != b.len)
return false;
while(a[tot] == b.a[tot] && tot >= 0)
tot--;
if(tot < 0)
return true;
return false;
}
void BigNum::print() //输出大数
{
int i;
cout << a[len - 1];
for(i = len-2; i >= 0; i--){
cout.width(DLEN);
cout.fill('0');
cout << a[i];
}
cout << endl;
}
int BigNum::Size()
{
int t = a[len-1],cnt = 0;
while(t){ t /= 10; cnt++; }
cnt += (len-1)*4;
return cnt;
}
int BigNum::the_first()
{
int t = a[len-1];
while(t > 10){ t /= 10;}
return t;
}
int BigNum::the_last()
{
int t = a[0];
return t%10;
}
int BigNum::to_int()
{
int i,num;
num = a[len-1];
for(i = len-2; i >= 0; i--)
num = num*(MAXN+1) + a[i];
return num;
}
long long int BigNum::to_long()
{
int i;
long long int num;
num = a[len-1];
for(i = len-2; i >= 0; i--)
num = num*(MAXN+1) + a[i];
return num;
}
string to_string2(int n)
{
ostringstream stream;
stream<<n; //n为int类型
return stream.str();
}
string BigNum::to_String()
{
int i;
string s = "",tp = "";
s += to_string2(a[len-1]);
for(i = len-2; i >= 0; i--){
tp = to_string2(a[i]);
int tot = tp.size();
tp.insert(tp.begin(),4-tot,'0');
s = s + tp;
}
return s;
}
BigNum Fibonacci(int n)
{
BigNum arr[2] = {0,1};
if(n <= 1)
return arr[n];
BigNum num; //用来保存计算结果
int i;
for(i = 2; i <= n; i++)
{
num = arr[1] + arr[0];
arr[0] = arr[1];
arr[1] = num;
}
return num;
}
int main()
{
BigNum a=1,b,c;
int n;
cin>>n;
for(int i = 1; i <= n; i++)
a=a*2;
//a.print();
c=Fibonacci(n+1);
//c.print();
b=a-c-c;
b.print();
}