计算长度为N的字母数字回文

给定正整数N ，任务是找到长度为N的字母数字回文字符串的数量。由于此类字符串的数量可能很大，因此请以10 ⁹ + 7为模输出答案。

例子：

Input: N = 2
Output: 62
Explanation: There are 26 palindromes of the form {“AA”, “BB”, …, “ZZ”}, 26 palindromes of the form {“aa”, “bb”, …, “cc”} and 10 palindromes of the form {“00”, “11”, …, “99”}. Therefore, the total number of palindromic strings = 26 + 26 + 10 = 62.

Input: N = 3
Output: 3844

天真的方法：最简单的方法是生成所有可能的长度为N的字母数字字符串，并针对每个字符串检查它是否是回文式。因为在每个位置上总共可以放置62个字符。因此，有62 ^N个可能的字符串。

时间复杂度： O(N * 62 ^N )
辅助空间： O(N)

高效的方法：为了优化上述方法，其思想是利用回文的性质。可以看出，如果字符串的长度是偶数，则对于从0到N / 2的每个索引i ，索引i和(N – 1 – i)处的字符都是相同的。因此，对于从0到N / 2的每个位置，都有62 ^{N / 2个}选项。同样，如果长度为奇数，则存在62 ^{(N + 1)/ 2个}选项。因此，可以说，对于一些N ，有62个^{ceil(N / 2)个}可能的回文字符串。
请按照以下步骤解决问题：

• 对于给定的N值，使用模幂计算62 ^{ceil(N / 2)} mod 10 ⁹ + 7 。
• 将62 ^{ceil(N / 2)} mod 10 ⁹ + 7打印为所需答案。

下面是上述方法的实现：

C++14

// C++ program for the
// above approach
#include 
using namespace std;
 
// Function to calculate
// (x ^ y) mod p
int power(int x, int y,
          int p)
{
  // Initialize result
  int res = 1;
 
  // Update x if it is more
  // than or equal to p
  x = x % p;
 
  if (x == 0)
    return 0;
 
  while (y > 0)
  {
    // If y is odd, multiply
    // x with result
    if ((y & 1) == 1)
      res = (res * x) % p;
 
    // y must be even now
    y = y >> 1;
    x = (x * x) % p;
  }
 
  // Return the final
  // result
  return res;
}
 
// Driver Code
int main()
{   
  // Given N
  int N = 3;
  int flag, k, m;
 
  // Base Case
  if((N == 1) ||
     (N == 2))
    cout << 62;
 
  else
    m = 1000000000 + 7;
 
  // Check whether n
  // is even or odd
  if(N % 2 == 0)
  {
    k = N / 2;
    flag = true;
  }
  else
  {
    k = (N - 1) / 2;
    flag = false;
  }
  if(flag != 0)
  {      
    // Function Call
    int a = power(62,
                  k, m);
    cout << a;
  }
  else
  {
    // Function Call
    int a = power(62,
                  (k + 1), m);
    cout << a;
  }
}
 
// This code is contributed by sanjoy_62

Java

// Java program for the
// above approach
import java.util.*;
class GFG{
 
// Function to calculate
// (x ^ y) mod p
static int power(int x,
                 int y, int p)
{
  // Initialize result
  int res = 1;
 
  // Update x if it is more
  // than or equal to p
  x = x % p;
 
  if (x == 0)
    return 0;
 
  while (y > 0)
  {
    // If y is odd, multiply
    // x with result
    if ((y & 1) == 1)
      res = (res * x) % p;
 
    // y must be even now
    y = y >> 1;
    x = (x * x) % p;
  }
 
  // Return the final
  // result
  return res;
}
 
// Driver Code
public static void main(String[] args)
{
  // Given N
  int N = 3;
  int flag, k, m =0;
 
  // Base Case
  if ((N == 1) || (N == 2))
    System.out.print(62);
  else
    m = 1000000000 + 7;
 
  // Check whether n
  // is even or odd
  if (N % 2 == 0)
  {
    k = N / 2;
    flag = 1;
  }
  else
  {
    k = (N - 1) / 2;
    flag = 0;
  }
   
  if (flag != 0)
  {
    // Function Call
    int a = power(62, k, m);
    System.out.print(a);
  }
  else
  {
    // Function Call
    int a = power(62, (k + 1), m);
    System.out.print(a);
  }
}
}
 
// This code is contributed by 29AjayKumar

Python3

# Python3 program for the above approach
 
# Function to calculate (x ^ y) mod p
def power(x, y, p):
 
    # Initialize result
    res = 1
 
    # Update x if it is more
    # than or equal to p
    x = x % p
 
    if (x == 0):
        return 0
 
    while (y > 0):
 
        # If y is odd, multiply
        # x with result
        if ((y & 1) == 1):
            res = (res * x) % p
 
        # y must be even now
        y = y >> 1
        x = (x * x) % p
 
    # Return the final result
    return res
 
 
# Driver Code
 
# Given N
N = 3
 
# Base Case
if((N == 1) or (N == 2)):
    print(62)
     
else:
 
    m = (10**9)+7
 
    # Check whether n
    # is even or odd
    if(N % 2 == 0):
        k = N//2
        flag = True
    else:
        k = (N - 1)//2
        flag = False
         
    if(flag):
       
        # Function Call
        a = power(62, k, m)
        print(a)
    else:
 
        # Function Call
        a = power(62, (k + 1), m)
        print(a)

C#

// C# program for the
// above approach
using System;
 
class GFG{
 
// Function to calculate
// (x ^ y) mod p
static int power(int x, int y,
                 int p)
{
   
  // Initialize result
  int res = 1;
 
  // Update x if it is more
  // than or equal to p
  x = x % p;
 
  if (x == 0)
    return 0;
 
  while (y > 0)
  {
     
    // If y is odd, multiply
    // x with result
    if ((y & 1) == 1)
      res = (res * x) % p;
 
    // y must be even now
    y = y >> 1;
    x = (x * x) % p;
  }
 
  // Return the final
  // result
  return res;
}
 
// Driver Code
public static void Main()
{
   
  // Given N
  int N = 3;
  int flag, k, m = 0;
 
  // Base Case
  if ((N == 1) || (N == 2))
    Console.Write(62);
  else
    m = 1000000000 + 7;
 
  // Check whether n
  // is even or odd
  if (N % 2 == 0)
  {
    k = N / 2;
    flag = 1;
  }
  else
  {
    k = (N - 1) / 2;
    flag = 0;
  }
   
  if (flag != 0)
  {
    // Function Call
    int a = power(62, k, m);
    Console.Write(a);
  }
  else
  {
     
    // Function Call
    int a = power(62, (k + 1), m);
    Console.Write(a);
  }
}
}
 
// This code is contributed by code_hunt

Javascript

 
// JavaScript program to implement
// the above approach
 
// Function to calculate
// (x ^ y) mod p
function power(x, y, p)
{
  // Initialize result
  let res = 1;
  
  // Update x if it is more
  // than or equal to p
  x = x % p;
  
  if (x == 0)
    return 0;
  
  while (y > 0)
  {
    // If y is odd, multiply
    // x with result
    if ((y & 1) == 1)
      res = (res * x) % p;
  
    // y must be even now
    y = y >> 1;
    x = (x * x) % p;
  }
  
  // Return the final
  // result
  return res;
}
// Driver Code
 
      // Given N
  let N = 3;
  let flag, k, m =0;
  
  // Base Case
  if ((N == 1) || (N == 2))
    document.write(62);
  else
    m = 1000000000 + 7;
  
  // Check whether n
  // is even or odd
  if (N % 2 == 0)
  {
    k = N / 2;
    flag = 1;
  }
  else
  {
    k = (N - 1) / 2;
    flag = 0;
  }
    
  if (flag != 0)
  {
    // Function Call
    let a = power(62, k, m);
    document.write(a);
  }
  else
  {
    // Function Call
    let a = power(62, (k + 1), m);
    document.write(a);
  }
           

输出：

3844

时间复杂度： O(log N)
辅助空间： O(N)