[蓝桥杯2015决赛]机器人繁殖
题目描述
X星系的机器人可以自动复制自己。它们用1年的时间可以复制出2个自己，然后就失去复制能力。
每年X星系都会选出1个新出生的机器人发往太空。也就是说，如果X星系原有机器人5个，
1年后总数是：5 + 9 = 14
2年后总数是：5 + 9 + 17 = 31
如果已经探测经过n年后的机器人总数s，你能算出最初有多少机器人吗？
输入
输入存在多组测试数据
对于每组测试数据，输入一行两个数字n和s，用空格分开，含义如上。n不大于100，s位数不超过50位。
输出
对于每组测试数据，要求输出一行，一个整数，表示最初有机器人多少个。
样例输入
| 2 31 97 2218388550399401452619230609499 |
样例输出
| 5 8 |
分析
假设初始的机器人数目为 a 0 a_0 a0, 第 n n n 年后新生（减去送去太空）的机器人为 a n a_n an, 那么
a 0 = a 0 a_0 = a_0 a0=a0
a 1 = 2 ∗ a 0 − 1 a_1 = 2 * a_0 -1 a1=2∗a0−1
a 2 = 2 ∗ a 1 − 1 a_2 = 2 * a_1 - 1 a2=2∗a1−1
… \quad\quad …
a n = 2 ∗ a n − 1 − 1 a_n = 2 * a_{n-1}- 1 an=2∗an−1−1
令 a n − t = 2 ∗ ( a n − 1 − t ) a_n - t = 2 * (a_{n-1} - t) an−t=2∗(an−1−t)
则 a n = 2 ∗ a n − 1 − t a_n = 2 * a_{n-1} - t an=2∗an−1−t, 所以 t = 1 t = 1 t=1
a 0 = a 0 a_0 = a_0 a0=a0
a 1 − 1 = 2 ∗ ( a 0 − 1 ) a_1 - 1 = 2 * (a_0 - 1) a1−1=2∗(a0−1)
a 2 − 1 = 2 ∗ ( a 1 − 1 ) a_2 - 1 = 2 * (a_1 - 1) a2−1=2∗(a1−1)
… \quad\quad …
a n − 1 = 2 ∗ ( a n − 1 − 1 ) a_n - 1 = 2 *(a_{n-1} - 1) an−1=2∗(an−1−1)
以 a 1 − 1 a_1 - 1 a1−1 为首项， 2 2 2为公比，构造等比数列
则 a 1 + a 2 + . . . + a n − n = 2 ∗ ( a 0 − 1 ) ∗ ( 2 n − 1 ) a_1 + a_2 + ... + a_n - n = 2 * (a_0 -1) * (2^n - 1) a1+a2+...+an−n=2∗(a0−1)∗(2n−1)
则 s = 2 ∗ ( a 0 − 1 ) ∗ ( 2 n − 1 ) + n + a 0 s = 2 * (a_0 -1) * (2^n - 1) + n + a_0 s=2∗(a0−1)∗(2n−1)+n+a0
a 0 = s + 2 n + 1 − 2 − n 2 n + 1 − 1 a_0 = \frac{s + 2^{n+1} - 2 - n}{2 ^ {n+1} - 1} a0=2n+1−1s+2n+1−2−n
C++直接调用大数板子
# include <iostream>
# include <vector>
# include <sstream>
# include <iomanip>
using namespace std;
const int base = 1000000000;
const int base_digits = 9;
struct Int {
vector<int> a;
int sign;
void trim() {
while (!a.empty() && !a.back())
a.pop_back();
if (a.empty())
sign = 1;
}
Int() : sign(1) {
};
Int(long long v) {
*this = v;
}
Int(const string &s) {
read(s);
}
int size(){
if(a.empty()) return 0;
int ans=(a.size()-1)*base_digits;
int ca=a.back();
while(ca)
ans++,ca/=10;
return ans;
}
Int operator ^(const Int &v){
Int ans=1,a=*this,b=v;
while(!b.isZero()){
if(b%2)
ans*=a;
a*=a,b/=2;
}
return ans;
}
string to_string(){
stringstream ss;
ss << *this;
string s;
ss >> s;
return s;
}
long long longValue() const {
long long res = 0;
for (int i = a.size() - 1; i >= 0; i--)
res = res * base + a[i];
return res * sign;
}
int sumof(){
string s = to_string();
int ans = 0;
for(auto c : s) ans += c - '0';
return ans;
}
void operator=(const Int &v) {
sign = v.sign;
a = v.a;
}
void operator=(long long v) {
sign = 1;
a.clear();
if (v < 0)
sign = -1, v = -v;
for (; v > 0; v = v / base)
a.push_back(v % base);
}
Int operator+(const Int &v) const {
if (sign == v.sign) {
Int res = v;
for (int i = 0, carry = 0; i < (int) max(a.size(), v.a.size()) || carry; ++i) {
if (i == (int) res.a.size())
res.a.push_back(0);
res.a[i] += carry + (i < (int) a.size() ? a[i] : 0);
carry = res.a[i] >= base;
if (carry)
res.a[i] -= base;
}
return res;
}
return *this - (-v);
}
Int operator-(const Int &v) const {
if (sign == v.sign) {
if (abs() >= v.abs()) {
Int res = *this;
for (int i = 0, carry = 0; i < (int) v.a.size() || carry; ++i) {
res.a[i] -= carry + (i < (int) v.a.size() ? v.a[i] : 0);
carry = res.a[i] < 0;
if (carry)
res.a[i] += base;
}
res.trim();
return res;
}
return -(v - *this);
}
return *this + (-v);
}
void operator*=(int v) {
if (v < 0)
sign = -sign, v = -v;
for (int i = 0, carry = 0; i < (int) a.size() || carry; ++i) {
if (i == (int) a.size())
a.push_back(0);
long long cur = a[i] * (long long) v + carry;
carry = (int) (cur / base);
a[i] = (int) (cur % base);
//asm("divl %%ecx" : "=a"(carry), "=d"(a[i]) : "A"(cur), "c"(base));
}
trim();
}
Int operator*(int v) const {
Int res = *this;
res *= v;
return res;
}
void operator*=(long long v) {
if (v < 0)
sign = -sign, v = -v;
for (int i = 0, carry = 0; i < (int) a.size() || carry; ++i) {
if (i == (int) a.size())
a.push_back(0);
long long cur = a[i] * (long long) v + carry;
carry = (int) (cur / base);
a[i] = (int) (cur % base);
//asm("divl %%ecx" : "=a"(carry), "=d"(a[i]) : "A"(cur), "c"(base));
}
trim();
}
Int operator*(long long v) const {
Int res = *this;
res *= v;
return res;
}
friend pair<Int, Int> divmod(const Int &a1, const Int &b1) {
int norm = base / (b1.a.back() + 1);
Int a = a1.abs() * norm;
Int b = b1.abs() * norm;
Int q, r;
q.a.resize(a.a.size());
for (int i = a.a.size() - 1; i >= 0; i--) {
r *= base;
r += a.a[i];
int s1 = r.a.size() <= b.a.size() ? 0 : r.a[b.a.size()];
int s2 = r.a.size() <= b.a.size() - 1 ? 0 : r.a[b.a.size() - 1];
int d = ((long long) base * s1 + s2) / b.a.back();
r -= b * d;
while (r < 0)
r += b, --d;
q.a[i] = d;
}
q.sign = a1.sign * b1.sign;
r.sign = a1.sign;
q.trim();
r.trim();
return make_pair(q, r / norm);
}
Int operator/(const Int &v) const {
return divmod(*this, v).first;
}
Int operator%(const Int &v) const {
return divmod(*this, v).second;
}
void operator/=(int v) {
if (v < 0)
sign = -sign, v = -v;
for (int i = (int) a.size() - 1, rem = 0; i >= 0; --i) {
long long cur = a[i] + rem * (long long) base;
a[i] = (int) (cur / v);
rem = (int) (cur % v);
}
trim();
}
Int operator/(int v) const {
Int res = *this;
res /= v;
return res;
}
int operator%(int v) const {
if (v < 0)
v = -v;
int m = 0;
for (int i = a.size() - 1; i >= 0; --i)
m = (a[i] + m * (long long) base) % v;
return m * sign;
}
void operator+=(const Int &v) {
*this = *this + v;
}
void operator-=(const Int &v) {
*this = *this - v;
}
void operator*=(const Int &v) {
*this = *this * v;
}
void operator/=(const Int &v) {
*this = *this / v;
}
Int operator ++(){
*this += 1;
return *this;
}
Int operator ++(int){
Int temp = *this;
*this += 1;
return temp;
}
Int operator --(){
*this -= 1;
return *this;
}
Int operator --(int){
Int temp = *this;
*this -= 1;
return temp;
}
bool operator<(const Int &v) const {
if (sign != v.sign)
return sign < v.sign;
if (a.size() != v.a.size())
return a.size() * sign < v.a.size() * v.sign;
for (int i = a.size() - 1; i >= 0; i--)
if (a[i] != v.a[i])
return a[i] * sign < v.a[i] * sign;
return false;
}
bool operator>(const Int &v) const {
return v < *this;
}
bool operator<=(const Int &v) const {
return !(v < *this);
}
bool operator>=(const Int &v) const {
return !(*this < v);
}
bool operator==(const Int &v) const {
return !(*this < v) && !(v < *this);
}
bool operator!=(const Int &v) const {
return *this < v || v < *this;
}
bool isZero() const {
return a.empty() || (a.size() == 1 && !a[0]);
}
Int operator-() const {
Int res = *this;
res.sign = -sign;
return res;
}
Int abs() const {
Int res = *this;
res.sign *= res.sign;
return res;
}
friend Int gcd(const Int &a, const Int &b) {
return b.isZero() ? a : gcd(b, a % b);
}
friend Int lcm(const Int &a, const Int &b) {
return a / gcd(a, b) * b;
}
void read(const string &s) {
sign = 1;
a.clear();
int pos = 0;
while (pos < (int) s.size() && (s[pos] == '-' || s[pos] == '+')) {
if (s[pos] == '-')
sign = -sign;
++pos;
}
for (int i = s.size() - 1; i >= pos; i -= base_digits) {
int x = 0;
for (int j = max(pos, i - base_digits + 1); j <= i; j++)
x = x * 10 + s[j] - '0';
a.push_back(x);
}
trim();
}
friend istream& operator>>(istream &stream, Int &v) {
string s;
stream >> s;
v.read(s);
return stream;
}
friend ostream& operator<<(ostream &stream, const Int &v) {
if (v.sign == -1)
stream << '-';
stream << (v.a.empty() ? 0 : v.a.back());
for (int i = (int) v.a.size() - 2; i >= 0; --i)
stream << setw(base_digits) << setfill('0') << v.a[i];
return stream;
}
static vector<int> convert_base(const vector<int> &a, int old_digits, int new_digits) {
vector<long long> p(max(old_digits, new_digits) + 1);
p[0] = 1;
for (int i = 1; i < (int) p.size(); i++)
p[i] = p[i - 1] * 10;
vector<int> res;
long long cur = 0;
int cur_digits = 0;
for (int i = 0; i < (int) a.size(); i++) {
cur += a[i] * p[cur_digits];
cur_digits += old_digits;
while (cur_digits >= new_digits) {
res.push_back(int(cur % p[new_digits]));
cur /= p[new_digits];
cur_digits -= new_digits;
}
}
res.push_back((int) cur);
while (!res.empty() && !res.back())
res.pop_back();
return res;
}
typedef vector<long long> vll;
static vll karatsubaMultiply(const vll &a, const vll &b) {
int n = a.size();
vll res(n + n);
if (n <= 32) {
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
res[i + j] += a[i] * b[j];
return res;
}
int k = n >> 1;
vll a1(a.begin(), a.begin() + k);
vll a2(a.begin() + k, a.end());
vll b1(b.begin(), b.begin() + k);
vll b2(b.begin() + k, b.end());
vll a1b1 = karatsubaMultiply(a1, b1);
vll a2b2 = karatsubaMultiply(a2, b2);
for (int i = 0; i < k; i++)
a2[i] += a1[i];
for (int i = 0; i < k; i++)
b2[i] += b1[i];
vll r = karatsubaMultiply(a2, b2);
for (int i = 0; i < (int) a1b1.size(); i++)
r[i] -= a1b1[i];
for (int i = 0; i < (int) a2b2.size(); i++)
r[i] -= a2b2[i];
for (int i = 0; i < (int) r.size(); i++)
res[i + k] += r[i];
for (int i = 0; i < (int) a1b1.size(); i++)
res[i] += a1b1[i];
for (int i = 0; i < (int) a2b2.size(); i++)
res[i + n] += a2b2[i];
return res;
}
Int operator*(const Int &v) const {
vector<int> a6 = convert_base(this->a, base_digits, 6);
vector<int> b6 = convert_base(v.a, base_digits, 6);
vll a(a6.begin(), a6.end());
vll b(b6.begin(), b6.end());
while (a.size() < b.size())
a.push_back(0);
while (b.size() < a.size())
b.push_back(0);
while (a.size() & (a.size() - 1))
a.push_back(0), b.push_back(0);
vll c = karatsubaMultiply(a, b);
Int res;
res.sign = sign * v.sign;
for (int i = 0, carry = 0; i < (int) c.size(); i++) {
long long cur = c[i] + carry;
res.a.push_back((int) (cur % 1000000));
carry = (int) (cur / 1000000);
}
res.a = convert_base(res.a, 6, base_digits);
res.trim();
return res;
}
friend Int max(const Int &a,const Int &b){
if(a<b){
return a;
}
return b;
}
friend Int max(const Int &a,const int32_t &B){
Int b = B;
return max(a,b);
}
friend Int max(const Int &a,const int64_t &B){
Int b = B;
return max(a,b);
}
friend Int min(const Int &a,const Int &b){
if(a>b){
return b;
}
return a;
}
friend Int min(const Int &a,const int32_t &B){
Int b = B;
return min(a,b);
}
friend Int min(const Int &a,const int64_t &B){
Int b = B;
return min(a,b);
}
friend Int pow(const Int &a,const Int &b){
Int _c = 1;
Int _b = b;
Int _a = a;
while(_b != 0){
if(_b%2){
_c *= _a;
}
_a *= _a;
_b /= 2;
}
return _c;
}
friend Int pow(const Int &a,const int32_t &B){
Int b = B;
return pow(a,b);
}
friend Int pow(const Int &a,const int64_t &B){
Int b = B;
return pow(a,b);
}
friend Int sqrt(Int a) {
Int x0 = a, x1 = (a+1)/2;
while (x1 < x0) {
x0 = x1;
x1 = (x1+a/x1)/2;
}
return x0;
}
};
int main(){
Int n;
Int s;
while (std::cin >> n >> s) {
Int t = (s + pow(2, n+1) - 2 - n) / (pow(2, n+1) - 1);
std::cout << t << '\n';
}
return 0;
}
Java 调用 BigInteger 类
import java.math.BigInteger;
import java.util.*;
public class Main {
public static void main(String[] args) {
Scanner in = new Scanner(System.in);
while (in.hasNext()) {
int n = in.nextInt();
BigInteger s = in.nextBigInteger();
BigInteger t = BigInteger.valueOf(2).pow(n+1);
BigInteger a = s.add(t).subtract(BigInteger.valueOf(2 + n)).divide(t.subtract(BigInteger.valueOf(1)));
System.out.println(a);
}
in.close();
}
}
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