# Find the n-th number whose binary representation is a palindrome

Last Updated : 24 May, 2024

Find the nth number whose binary representation is a palindrome. Do not consider the leading zeros, while considering the binary representation. Consider the 1st number whose binary representation is a palindrome as 1, instead of 0Â

Examples:Â

`Input : 1Output : 11st Number whose binary representation is palindrome is 1 (1)Input : 9Output : 279th Number whose binary representationis palindrome is 27 (11011)`

Method 1: Naive

A naive approach would be, to traverse through all the integers from 1 to 2^31 – 1 and increment the palindrome count, if the number is a palindrome. When the palindrome count reaches the required n, break the loop and return the current integer.Â

C++ ```// C++ program to find n-th number whose binary // representation is palindrome. #include <bits/stdc++.h> using namespace std; // Finds if the kth bit is set in the binary // representation int isKthBitSet(int x, int k) { return (x & (1 << (k - 1))) ? 1 : 0; } // Returns the position of leftmost set bit // in the binary representation int leftmostSetBit(int x) { int count = 0; while (x) { count++; x = x >> 1; } return count; } // Finds whether the integer in binary // representation is palindrome or not int isBinPalindrome(int x) { int l = leftmostSetBit(x); int r = 1; // One by one compare bits while (l > r) { // Compare left and right bits and converge if (isKthBitSet(x, l) != isKthBitSet(x, r)) return 0; l--; r++; } return 1; } int findNthPalindrome(int n) { int pal_count = 0; // Start from 1, traverse through // all the integers int i = 0; for (i = 1; i <= INT_MAX; i++) { if (isBinPalindrome(i)) { pal_count++; } // If we reach n, break the loop if (pal_count == n) break; } return i; } // Driver code int main() { int n = 9; // Function Call cout << findNthPalindrome(n); } // This code is contributed // by Akanksha Rai ``` C ```// C program to find n-th number whose binary // representation is palindrome. #include <stdio.h> #define INT_MAX 2147483647 // Finds if the kth bit is set in the binary // representation int isKthBitSet(int x, int k) { return (x & (1 << (k - 1))) ? 1 : 0; } // Returns the position of leftmost set bit // in the binary representation int leftmostSetBit(int x) { int count = 0; while (x) { count++; x = x >> 1; } return count; } // Finds whether the integer in binary // representation is palindrome or not int isBinPalindrome(int x) { int l = leftmostSetBit(x); int r = 1; // One by one compare bits while (l > r) { // Compare left and right bits and converge if (isKthBitSet(x, l) != isKthBitSet(x, r)) return 0; l--; r++; } return 1; } int findNthPalindrome(int n) { int pal_count = 0; // Start from 1, traverse through // all the integers int i = 0; for (i = 1; i <= INT_MAX; i++) { if (isBinPalindrome(i)) { pal_count++; } // If we reach n, break the loop if (pal_count == n) break; } return i; } // Driver code int main() { int n = 9; // Function Call printf("%d", findNthPalindrome(n)); } ``` Java ```// Java program to find n-th // number whose binary // representation is palindrome. import java.io.*; class GFG { static int INT_MAX = 2147483647; // Finds if the kth bit // is set in the binary // representation static int isKthBitSet(int x, int k) { return ((x & (1 << (k - 1))) > 0) ? 1 : 0; } // Returns the position of // leftmost set bit in the // binary representation static int leftmostSetBit(int x) { int count = 0; while (x > 0) { count++; x = x >> 1; } return count; } // Finds whether the integer // in binary representation is // palindrome or not static int isBinPalindrome(int x) { int l = leftmostSetBit(x); int r = 1; // One by one compare bits while (l > r) { // Compare left and right // bits and converge if (isKthBitSet(x, l) != isKthBitSet(x, r)) return 0; l--; r++; } return 1; } static int findNthPalindrome(int n) { int pal_count = 0; // Start from 1, traverse // through all the integers int i = 0; for (i = 1; i <= INT_MAX; i++) { if (isBinPalindrome(i) > 0) { pal_count++; } // If we reach n, // break the loop if (pal_count == n) break; } return i; } // Driver code public static void main(String[] args) { int n = 9; // Function Call System.out.println(findNthPalindrome(n)); } } // This code is contributed // by anuj_67. ``` Python ```# Python 3 program to find n-th number # whose binary representation is palindrome. INT_MAX = 2147483647 # Finds if the kth bit is set in # the binary representation def isKthBitSet(x, k): return 1 if (x & (1 << (k - 1))) else 0 # Returns the position of leftmost # set bit in the binary representation def leftmostSetBit(x): count = 0 while (x): count += 1 x = x >> 1 return count # Finds whether the integer in binary # representation is palindrome or not def isBinPalindrome(x): l = leftmostSetBit(x) r = 1 # One by one compare bits while (l > r): # Compare left and right bits # and converge if (isKthBitSet(x, l) != isKthBitSet(x, r)): return 0 l -= 1 r += 1 return 1 def findNthPalindrome(n): pal_count = 0 # Start from 1, traverse # through all the integers i = 0 for i in range(1, INT_MAX + 1): if (isBinPalindrome(i)): pal_count += 1 # If we reach n, break the loop if (pal_count == n): break return i # Driver code if __name__ == "__main__": n = 9 # Function Call print(findNthPalindrome(n)) # This code is contributed # by ChitraNayal ``` C# ```// C# program to find n-th // number whose binary // representation is palindrome. using System; class GFG { static int INT_MAX = 2147483647; // Finds if the kth bit // is set in the binary // representation static int isKthBitSet(int x, int k) { return ((x & (1 << (k - 1))) > 0) ? 1 : 0; } // Returns the position of // leftmost set bit in the // binary representation static int leftmostSetBit(int x) { int count = 0; while (x > 0) { count++; x = x >> 1; } return count; } // Finds whether the integer // in binary representation is // palindrome or not static int isBinPalindrome(int x) { int l = leftmostSetBit(x); int r = 1; // One by one compare bits while (l > r) { // Compare left and right // bits and converge if (isKthBitSet(x, l) != isKthBitSet(x, r)) return 0; l--; r++; } return 1; } static int findNthPalindrome(int n) { int pal_count = 0; // Start from 1, traverse // through all the integers int i = 0; for (i = 1; i <= INT_MAX; i++) { if (isBinPalindrome(i) > 0) { pal_count++; } // If we reach n, // break the loop if (pal_count == n) break; } return i; } // Driver code static public void Main() { int n = 9; // Function Call Console.WriteLine(findNthPalindrome(n)); } } // This code is contributed ajit ``` Javascript ```// Javascript program to find n-th // number whose binary // representation is palindrome. let INT_MAX = 2147483647; // Finds if the kth bit // is set in the binary // representation function isKthBitSet(x, k) { return ((x & (1 << (k - 1))) > 0) ? 1 : 0; } // Returns the position of // leftmost set bit in the // binary representation function leftmostSetBit(x) { let count = 0; while (x > 0) { count++; x = x >> 1; } return count; } // Finds whether the integer // in binary representation is // palindrome or not function isBinPalindrome(x) { let l = leftmostSetBit(x); let r = 1; // One by one compare bits while (l > r) { // Compare left and right // bits and converge if (isKthBitSet(x, l) != isKthBitSet(x, r)) return 0; l--; r++; } return 1; } function findNthPalindrome(n) { let pal_count = 0; // Start from 1, traverse // through all the integers let i = 0; for(i = 1; i <= INT_MAX; i++) { if (isBinPalindrome(i) > 0) { pal_count++; } // If we reach n, // break the loop if (pal_count == n) break; } return i; } // Driver code let n = 9; // Function Call console.log(findNthPalindrome(n)); // This code is contributed by suresh07 ``` PHP ```<?php // PHP program to find n-th number whose // binary representation is palindrome. // Finds if the kth bit is set in // the binary representation function isKthBitSet(\$x, \$k) { return (\$x & (1 << (\$k - 1))) ? 1 : 0; } // Returns the position of leftmost set // bit in the binary representation function leftmostSetBit(\$x) { \$count = 0; while (\$x) { \$count++; \$x = \$x >> 1; } return \$count; } // Finds whether the integer in binary // representation is palindrome or not function isBinPalindrome(\$x) { \$l = leftmostSetBit(\$x); \$r = 1; // One by one compare bits while (\$l > \$r) { // Compare left and right bits // and converge if (isKthBitSet(\$x, \$l) != isKthBitSet(\$x, \$r)) return 0; \$l--; \$r++; } return 1; } function findNthPalindrome(\$n) { \$pal_count = 0; // Start from 1, traverse through // all the integers \$i = 0; for (\$i = 1; \$i <= PHP_INT_MAX; \$i++) { if (isBinPalindrome(\$i)) { \$pal_count++; } // If we reach n, break the loop if (\$pal_count == \$n) break; } return \$i; } // Driver code \$n = 9; // Function Call echo (findNthPalindrome(\$n)); // This code is contributed by jit_t ?> ```

Output
`27`

Time complexity: O(x) where x is a resultant number.Â
Auxiliary Space: O(1), the space complexity is O(1) since we are only using a constant amount of memory to store the values.

Method 2: Using BFS

In this approach first, we simply add Â  “11” this string into the queue. And then for every string, we have two cases. i.eÂ

1. if the curr string of even length then add “0” and “1” at the mid of curr string and add it into the queue.
2. if the curr string is of odd length then add mid char of the curr string into the resultant string and then add it into the queue.

if curr string is of even length

if curr string is of odd length

Â Below is the implementation of the above approach:

C++ ```// C++ program to find n-th palindrome #include <bits/stdc++.h> using namespace std; // utility function which is used to // convert binary string into integer int convertStringToInt(string s) { int num = 0; // convert binary string into integer for (int i = 0; i < s.size(); i++) { num = num * 2 + (s[i] - '0'); } return num; } // function to find nth binary palindrome number int getNthNumber(int n) { // base case if (n == 1) return 1; n--; // stores the binary palindrome string queue<string> q; // add 2nd binary palindrome string q.push("11"); // runs till the nth binary palindrome number while (!q.empty()) { // remove curr binary palindrome string from queue string curr = q.front(); q.pop(); n--; // if n==0 then we find the n'th binary palindrome // so we return our answer if (!n) { return convertStringToInt(curr); } int mid = curr.size() / 2; // if length is even .so it is our first case // we have two choices if (curr.size() % 2 == 0) { string s0 = curr, s1 = curr; s0.insert(mid, "0"); s1.insert(mid, "1"); q.push(s0); q.push(s1); } // if length is odd .so it is our second case // we have only one choice else { string ch(1, curr[mid]); string temp = curr; temp.insert(mid, ch); q.push(temp); } } return 0; } // Driver Code int main() { int n = 9; // Function Call cout << getNthNumber(n); } // This code is contributed by Sagar Jangra and Naresh // Saharan ``` Java ```// Java program to find n-th palindrome import java.io.*; import java.util.*; class GFG { // utility function which is used to // convert binary string into integer public static int convertStringToInt(String s) { int ans = 0; // convert binary string into integer for (int i = 0; i < s.length(); i++) { if (s.charAt(i) == '1') ans += 1 << i; } return ans; } // function to find nth binary palindrome number public static int getNthNumber(int n) { // stores the binary palindrome string Queue<String> q = new LinkedList<>(); // base case if (n == 1) return 1; n = n - 1; // add 2nd binary palindrome string q.add("11"); // runs till the nth binary palindrome number while (n-- > 0) { // remove curr binary palindrome string from // queue String curr = q.remove(); // if n==0 then we find the n'th binary // palindrome so we return our answer if (n == 0) return convertStringToInt(curr); // calculate length of curr binary palindrome // string int len = curr.length(); // if length is even .so it is our first case // we have two choices if (len % 2 == 0) { q.add(curr.substring(0, len / 2) + "0" + curr.substring(len / 2)); q.add(curr.substring(0, len / 2) + "1" + curr.substring(len / 2)); } // if length is odd .so it is our second case // we have only one choice else { char midChar = curr.charAt(len / 2); q.add(curr.substring(0, len / 2) + midChar + curr.substring(len / 2)); } } return -1; } // Driver code public static void main(String[] args) { int n = 9; // Function Call System.out.println(getNthNumber(n)); } } // This code is contributed by Naresh Saharan and Sagar // Jangra ``` Python ```# Python program to find n-th palindrome # utility function which is used to # convert binary string into integer def convertStringToInt(s): ans = 0 # convert binary string into integer for i in range(len(s)): ans = ans * 2 + (ord(s[i]) - ord('0')) return ans # function to find nth binary palindrome number def getNthNumber(n): # stores the binary palindrome string q = [] # base case if(n == 1): return 1 n = n - 1 # add 2nd binary palindrome string q.append("11") # runs till the nth binary palindrome number while(len(q) != 0): # remove curr binary palindrome string from # queue curr = q.pop(0) n -= 1 # if n==0 then we find the n'th binary # palindrome so we return our answer if(n ==0): return convertStringToInt(curr) # calculate length of curr binary palindrome # string lenn = len(curr) # if length is even .so it is our first case # we have two choices if (len(curr) % 2 == 0): q.append(curr[0:int(lenn/2)]+"0"+curr[int(lenn/2):]) q.append(curr[0:int(lenn/2)]+"1"+curr[int(lenn/2):]) # if length is odd .so it is our second case # we have only one choice else: midChar = curr[int(lenn/2)] q.append(curr[0:int(lenn/2)]+midChar+curr[int(lenn/2):]) return 0 # Driver code n = 9 # Function Call print(getNthNumber(n)) # This code is contributed by avanitrachhadiya2155 ``` C# ```// C# program to find n-th palindrome using System; using System.Collections.Generic; class GFG{ // Utility function which is used to // convert binary string into integer public static int convertStringToInt(String s) { int ans = 0; // Convert binary string into integer for(int i = 0; i < s.Length; i++) { if (s[i] == '1') ans += 1 << i; } return ans; } // Function to find nth binary palindrome number public static int getNthNumber(int n) { // Stores the binary palindrome string Queue<String> q = new Queue<String>(); // Base case if (n == 1) return 1; n = n - 1; // Add 2nd binary palindrome string q.Enqueue("11"); // Runs till the nth binary palindrome number while (n-- > 0) { // Remove curr binary palindrome // string from queue String curr = q.Dequeue(); // If n==0 then we find the n'th binary // palindrome so we return our answer if (n == 0) return convertStringToInt(curr); // Calculate length of curr binary palindrome // string int len = curr.Length; // If length is even .so it is our first case // we have two choices if (len % 2 == 0) { q.Enqueue(curr.Substring(0, len / 2) + "0" + curr.Substring(len / 2)); q.Enqueue(curr.Substring(0, len / 2) + "1" + curr.Substring(len / 2)); } // If length is odd .so it is our second case // we have only one choice else { char midChar = curr[len / 2]; q.Enqueue(curr.Substring(0, len / 2) + midChar + curr.Substring(len / 2)); } } return -1; } // Driver code public static void Main(String[] args) { int n = 9; // Function Call Console.WriteLine(getNthNumber(n)); } } // This code is contributed by 29AjayKumar ``` Javascript ```// Javascript program to find n-th palindrome // utility function which is used to // convert binary string into integer function convertStringToInt(s) { let ans = 0; // convert binary string into integer for (let i = 0; i < s.length; i++) { if (s[i] == '1') ans += 1 << i; } return ans; } // function to find nth binary palindrome number function getNthNumber(n) { // stores the binary palindrome string let q = []; // base case if (n == 1) return 1; n = n - 1; // add 2nd binary palindrome string q.push("11"); // runs till the nth binary palindrome number while (n-- > 0) { // remove curr binary palindrome string from // queue let curr = q.shift(); // if n==0 then we find the n'th binary // palindrome so we return our answer if (n == 0) return convertStringToInt(curr); // calculate length of curr binary palindrome // string let len = curr.length; let len2 = Math.floor(len / 2); // if length is even .so it is our first case // we have two choices if (len % 2 == 0) { q.push(curr.substring(0, len2) + "0" + curr.substring(len2)); q.push(curr.substring(0, len2) + "1" + curr.substring(len2)); } // if length is odd .so it is our second case // we have only one choice else { let midChar = curr[len2]; q.push(curr.substring(0, len2) + midChar + curr.substring(len2)); } } return -1; } // Driver code let n = 9; // Function Call console.log(getNthNumber(n)); // This code is contributed by ab2127 ```

Output
`27`

Time complexity: O(N)
Auxiliary Space: O(N)

Method 3: Constructing the nth palindrome

We can construct the nth binary palindrome in its binary representation directly using the below approach.Â
If we observe the first few binary palindromesÂ

` *         | nth Binary  |       n   | Palindrome  |     Group           |             |--------------------------- Group 0    1 --->  1 (1)    Group 1 (Will have binary representation of length 2*(1)and 2*(1) + 1)    Fix the first and last bit as 1 and insert nothing    (|) in between. Length is 2*(1)    2 --->  1|1 (3)         Fix the first and last bit as 1 and insert bit 0    in between. Length is 2*(1) + 1    3 --->  101 (5)        Fix the first and last bit as 1 and insert bit 1     in between. Length is 2*(1) + 1     4 --->  111 (7)        FGroup 2 (Will have binary representation of length     2*(2) and 2*(2) + 1).  Fix the first and last     bit as 1 and insert nothing (|) at middle.     And put 0 in binary format in both directions     from middle. Length is 2*(2)    5 --->  10|01           Fix the first and last bit as 1 and insert     nothing (|) at middle. And put 1 in binary     format in both directions from middle.     Length is 2*(2)    6 --->  11|11              7 --->  10001          8 --->  10101         9 --->  11011         10 --->  11111          Group 3 (Will have binary representation of         length 2*(3) and 2*(3) + 1)    11 ---> 100|001        12 ---> 101|101        13 ---> 110|011        14 ---> 111|111          15 ---> 1000001      16 ---> 1001001      17 ---> 1010101      18 ---> 1011101      19 ---> 1100011      20 ---> 1101011      21 ---> 1110111        22 ---> 1111111 -------------------- `

Algorithm:Â
1) We can divide the set of palindrome numbers into some groups.Â
2) n-th group will have (2^(n-1) + 2^n = 3 * 2 ^(n-1) ) number of binary palindromesÂ
3) With the given number, we can find the group to which it belongs and the offset in that group.Â
4) As the leading zeros are not to be considered, we should use bit 1 as the starting bit and the ending bit of the number in binary representationÂ
5) And we will fill other bits based on the groupno and groupoffsetÂ
6) Based on the offset, we can find which bit should be inserted at the middle (|(nothing) or 0 or 1) andÂ
which number(in binary form) (1 or 2 or 3 or 4 or ..) should be placed in both directions from the middle

Consider Below ExampleÂ

`Let us construct the 8th binary palindrome numberThe group number will be 2, and no.of elements before that group are 1 + 3 * 2^1 which is 4So the offset for the 8th element will be 8 - 4 - 1 = 3And first 2^(groupno - 1) = 2^1, elements will have even length(in binary representation) of2*groupno, next 2^groupno elements will have odd length(in binary representation) of 2*groupno + 1Place bit 1 as the first bit and as the last bit (firstbit: 0, last bit: 2*groupno or 2*groupno - 1)As the offset is 3, 4th(3 + 1) element in thegroup, will have odd length and have 1 in the middleBelow is the table of middle bit to be used for the given offset for the group 2offset    middle bit  0            |  1            |  2            0  3            1  4            0  5            1And we should be filling the binary representation of number 0(((groupoffset) - 2^(groupno-1)) /2)from middle n both directions1 0 1 0 1FirstElement Number MiddleElement Number LastElement    1           0         1         0         1The 8th number will be 21`

Below is the implementation of the above idea :Â

C++ ```// Efficient C++ program to find n-th palindrome #include <iostream> #include <bits/stdc++.h> using namespace std; // Construct the nth binary palindrome with the // given group number, aux_number and operation // type int constructNthNumber(int group_no, int aux_num, int op) { int INT_SIZE = 32 ; int a[INT_SIZE] = { 0 }; int num = 0, len_f; int i = 0; // No need to insert any bit in the middle if (op == 2) { // Length of the final binary representation len_f = 2 * group_no; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; // Start filling the a[] from middle, // with the aux_num binary representation while (aux_num) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Insert bit 0 in the middle else if (op == 0) { // Length of the final binary representation len_f = 2 * group_no + 1; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; a[group_no] = 0; // Start filling the a[] from middle, with // the aux_num binary representation while (aux_num) { // Get the auxiliary number's ith bit // and fill around middle a[group_no + 1 + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Insert bit 1 in the middle else { // Length of the final binary representation len_f = 2 * group_no + 1; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; a[group_no] = 1; // Start filling the a[] from middle, with // the aux_num binary representation while (aux_num) { // Get the auxiliary number's ith bit // and fill around middle a[group_no + 1 + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Convert the number to decimal from binary for(i = 0; i < len_f; i++) num += (1 << i) * a[i]; return num; } // Will return the nth binary palindrome number int getNthNumber(int n) { int group_no = 0, group_offset; int count_upto_group = 0, count_temp = 1; int op, aux_num; // Add number of elements in all the groups, // until the group of the nth number is found while (count_temp < n) { group_no++; // Total number of elements until this group count_upto_group = count_temp; count_temp += 3 * (1 << (group_no - 1)); } // Element's offset position in the group group_offset = n - count_upto_group - 1; // Finding which bit to be placed in the // middle and finding the number, which we // will fill from the middle in both // directions if ((group_offset + 1) <= (1 << (group_no - 1))) { // No need to put extra bit in middle op = 2; // We need to fill this auxiliary number // in binary form the middle in both directions aux_num = group_offset; } else { if (((group_offset + 1) - (1 << (group_no - 1))) % 2) // Need to Insert 0 at middle op = 0; else // Need to Insert 1 at middle op = 1; aux_num = ((group_offset) - (1 << (group_no - 1))) / 2; } return constructNthNumber(group_no, aux_num, op); } // Driver code int main() { int n = 9; // Function Call cout << getNthNumber(n) ; return 0; } // This code is contributed by Khushboogoyal499 ``` C ```// Efficient C program to find n-th palindrome #include <stdio.h> #define INT_SIZE 32 // Construct the nth binary palindrome with the // given group number, aux_number and operation // type int constructNthNumber(int group_no, int aux_num, int op) { int a[INT_SIZE] = { 0 }; int num = 0, len_f; int i = 0; // No need to insert any bit in the middle if (op == 2) { // Length of the final binary representation len_f = 2 * group_no; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; // Start filling the a[] from middle, // with the aux_num binary representation while (aux_num) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Insert bit 0 in the middle else if (op == 0) { // Length of the final binary representation len_f = 2 * group_no + 1; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; a[group_no] = 0; // Start filling the a[] from middle, with // the aux_num binary representation while (aux_num) { // Get the auxiliary number's ith bit and fill // around middle a[group_no + 1 + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } else // Insert bit 1 in the middle { // Length of the final binary representation len_f = 2 * group_no + 1; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; a[group_no] = 1; // Start filling the a[] from middle, with // the aux_num binary representation while (aux_num) { // Get the auxiliary number's ith bit and fill // around middle a[group_no + 1 + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Convert the number to decimal from binary for (i = 0; i < len_f; i++) num += (1 << i) * a[i]; return num; } // Will return the nth binary palindrome number int getNthNumber(int n) { int group_no = 0, group_offset; int count_upto_group = 0, count_temp = 1; int op, aux_num; // Add number of elements in all the groups, // until the group of the nth number is found while (count_temp < n) { group_no++; // Total number of elements until this group count_upto_group = count_temp; count_temp += 3 * (1 << (group_no - 1)); } // Element's offset position in the group group_offset = n - count_upto_group - 1; // Finding which bit to be placed in the // middle and finding the number, which we // will fill from the middle in both // directions if ((group_offset + 1) <= (1 << (group_no - 1))) { op = 2; // No need to put extra bit in middle // We need to fill this auxiliary number // in binary form the middle in both directions aux_num = group_offset; } else { if (((group_offset + 1) - (1 << (group_no - 1))) % 2) op = 0; // Need to Insert 0 at middle else op = 1; // Need to Insert 1 at middle aux_num = ((group_offset) - (1 << (group_no - 1))) / 2; } return constructNthNumber(group_no, aux_num, op); } // Driver code int main() { int n = 9; // Function Call printf("%d", getNthNumber(n)); return 0; } ``` Java ```// Efficient Java program to find n-th palindrome class GFG { static int INT_SIZE = 32; // Construct the nth binary palindrome with the // given group number, aux_number and operation // type static int constructNthNumber(int group_no, int aux_num, int op) { int a[] = new int[INT_SIZE]; int num = 0, len_f; int i = 0; // No need to insert any bit in the middle if (op == 2) { // Length of the final binary representation len_f = 2 * group_no; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; // Start filling the a[] from middle, // with the aux_num binary representation while (aux_num > 0) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Insert bit 0 in the middle else if (op == 0) { // Length of the final binary representation len_f = 2 * group_no + 1; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; a[group_no] = 0; // Start filling the a[] from middle, with // the aux_num binary representation while (aux_num > 0) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + 1 + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } else // Insert bit 1 in the middle { // Length of the final binary representation len_f = 2 * group_no + 1; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; a[group_no] = 1; // Start filling the a[] from middle, with // the aux_num binary representation while (aux_num > 0) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + 1 + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Convert the number to decimal from binary for (i = 0; i < len_f; i++) num += (1 << i) * a[i]; return num; } // Will return the nth binary palindrome number static int getNthNumber(int n) { int group_no = 0, group_offset; int count_upto_group = 0, count_temp = 1; int op, aux_num; // Add number of elements in all the groups, // until the group of the nth number is found while (count_temp < n) { group_no++; // Total number of elements until this group count_upto_group = count_temp; count_temp += 3 * (1 << (group_no - 1)); } // Element's offset position in the group group_offset = n - count_upto_group - 1; // Finding which bit to be placed in the // middle and finding the number, which we // will fill from the middle in both // directions if ((group_offset + 1) <= (1 << (group_no - 1))) { op = 2; // No need to put extra bit in middle // We need to fill this auxiliary number // in binary form the middle in both directions aux_num = group_offset; } else { if (((group_offset + 1) - (1 << (group_no - 1))) % 2 == 1) op = 0; // Need to Insert 0 at middle else op = 1; // Need to Insert 1 at middle aux_num = ((group_offset) - (1 << (group_no - 1))) / 2; } return constructNthNumber(group_no, aux_num, op); } // Driver code public static void main(String[] args) { int n = 9; // Function Call System.out.printf("%d", getNthNumber(n)); } } /* This code contributed by PrinciRaj1992 */ ``` Python ```# Efficient Python3 program to find n-th palindrome INT_SIZE = 32 # Construct the nth binary palindrome with the # given group number, aux_number and operation type def constructNthNumber(group_no, aux_num, op): a = [0] * INT_SIZE num, i = 0, 0 # No need to insert any bit in the middle if op == 2: # Length of the final binary representation len_f = 2 * group_no # Fill first and last bit as 1 a[len_f - 1] = a[0] = 1 # Start filling the a[] from middle, # with the aux_num binary representation while aux_num: # Get the auxiliary number's ith # bit and fill around middle a[group_no + i] = a[group_no - 1 - i] = \ aux_num & 1 aux_num = aux_num >> 1 i += 1 # Insert bit 0 in the middle elif op == 0: # Length of the final binary representation len_f = 2 * group_no + 1 # Fill first and last bit as 1 a[len_f - 1] = a[0] = 1 a[group_no] = 0 # Start filling the a[] from middle, with # the aux_num binary representation while aux_num: # Get the auxiliary number's ith # bit and fill around middle a[group_no + 1 + i] = a[group_no - 1 - i] = \ aux_num & 1 aux_num = aux_num >> 1 i += 1 else: # Insert bit 1 in the middle # Length of the final binary representation len_f = 2 * group_no + 1 # Fill first and last bit as 1 a[len_f - 1] = a[0] = 1 a[group_no] = 1 # Start filling the a[] from middle, with # the aux_num binary representation while aux_num: # Get the auxiliary number's ith # bit and fill around middle a[group_no + 1 + i] = a[group_no - 1 - i] = \ aux_num & 1 aux_num = aux_num >> 1 i += 1 # Convert the number to decimal from binary for i in range(0, len_f): num += (1 << i) * a[i] return num # Will return the nth binary palindrome number def getNthNumber(n): group_no = 0 count_upto_group, count_temp = 0, 1 # Add number of elements in all the groups, # until the group of the nth number is found while count_temp < n: group_no += 1 # Total number of elements until this group count_upto_group = count_temp count_temp += 3 * (1 << (group_no - 1)) # Element's offset position in the group group_offset = n - count_upto_group - 1 # Finding which bit to be placed in the # middle and finding the number, which we # will fill from the middle in both directions if (group_offset + 1) <= (1 << (group_no - 1)): op = 2 # No need to put extra bit in middle # We need to fill this auxiliary number # in binary form the middle in both directions aux_num = group_offset else: if (((group_offset + 1) - (1 << (group_no - 1))) % 2): op = 0 # Need to Insert 0 at middle else: op = 1 # Need to Insert 1 at middle aux_num = (((group_offset) - (1 << (group_no - 1))) // 2) return constructNthNumber(group_no, aux_num, op) # Driver code if __name__ == "__main__": n = 9 # Function Call print(getNthNumber(n)) # This code is contributed by Rituraj Jain ``` C# ```// Efficient C# program to find n-th palindrome using System; class GFG { static int INT_SIZE = 32; // Construct the nth binary palindrome with the // given group number, aux_number and operation // type static int constructNthNumber(int group_no, int aux_num, int op) { int[] a = new int[INT_SIZE]; int num = 0, len_f; int i = 0; // No need to insert any bit in the middle if (op == 2) { // Length of the final binary representation len_f = 2 * group_no; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; // Start filling the a[] from middle, // with the aux_num binary representation while (aux_num > 0) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Insert bit 0 in the middle else if (op == 0) { // Length of the final binary representation len_f = 2 * group_no + 1; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; a[group_no] = 0; // Start filling the a[] from middle, with // the aux_num binary representation while (aux_num > 0) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + 1 + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } else // Insert bit 1 in the middle { // Length of the final binary representation len_f = 2 * group_no + 1; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; a[group_no] = 1; // Start filling the a[] from middle, with // the aux_num binary representation while (aux_num > 0) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + 1 + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Convert the number to decimal from binary for (i = 0; i < len_f; i++) num += (1 << i) * a[i]; return num; } // Will return the nth binary palindrome number static int getNthNumber(int n) { int group_no = 0, group_offset; int count_upto_group = 0, count_temp = 1; int op, aux_num; // Add number of elements in all the groups, // until the group of the nth number is found while (count_temp < n) { group_no++; // Total number of elements until this group count_upto_group = count_temp; count_temp += 3 * (1 << (group_no - 1)); } // Element's offset position in the group group_offset = n - count_upto_group - 1; // Finding which bit to be placed in the // middle and finding the number, which we // will fill from the middle in both // directions if ((group_offset + 1) <= (1 << (group_no - 1))) { op = 2; // No need to put extra bit in middle // We need to fill this auxiliary number // in binary form the middle in both directions aux_num = group_offset; } else { if (((group_offset + 1) - (1 << (group_no - 1))) % 2 == 1) op = 0; // Need to Insert 0 at middle else op = 1; // Need to Insert 1 at middle aux_num = ((group_offset) - (1 << (group_no - 1))) / 2; } return constructNthNumber(group_no, aux_num, op); } // Driver code public static void Main(String[] args) { int n = 9; // Function Call Console.Write("{0}", getNthNumber(n)); } } // This code contributed by Rajput-Ji ``` Javascript ```// Efficient javascript program to find n-th palindrome var INT_SIZE = 32; // Construct the nth binary palindrome with the // given group number, aux_number and operation // type function constructNthNumber(group_no , aux_num,op) { var a = Array.from({length: INT_SIZE}, (_, i) => 0); var num = 0, len_f=0; var i = 0; // No need to insert any bit in the middle if (op == 2) { // Length of the final binary representation len_f = 2 * group_no; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; // Start filling the a from middle, // with the aux_num binary representation while (aux_num > 0) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Insert bit 0 in the middle else if (op == 0) { // Length of the final binary representation len_f = 2 * group_no + 1; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; a[group_no] = 0; // Start filling the a from middle, with // the aux_num binary representation while (aux_num > 0) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + 1 + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Insert bit 1 in the middle else { // Length of the final binary representation len_f = 2 * group_no + 1; // Fill first and last bit as 1 a[len_f - 1] = a[0] = 1; a[group_no] = 1; // Start filling the a from middle, with // the aux_num binary representation while (aux_num > 0) { // Get the auxiliary number's ith bit and // fill around middle a[group_no + 1 + i] = a[group_no - 1 - i] = aux_num & 1; aux_num = aux_num >> 1; i++; } } // Convert the number to decimal from binary for(i = 0; i < len_f; i++) num += (1 << i) * a[i]; return num; } // Will return the nth binary palindrome number function getNthNumber(n) { var group_no = 0, group_offset; var count_upto_group = 0, count_temp = 1; var op, aux_num; // Add number of elements in all the groups, // until the group of the nth number is found while (count_temp < n) { group_no++; // Total number of elements until this group count_upto_group = count_temp; count_temp += 3 * (1 << (group_no - 1)); } // Element's offset position in the group group_offset = n - count_upto_group - 1; // Finding which bit to be placed in the // middle and finding the number, which we // will fill from the middle in both // directions if ((group_offset + 1) <= (1 << (group_no - 1))) { // No need to put extra bit in middle op = 2; // We need to fill this auxiliary number // in binary form the middle in both directions aux_num = group_offset; } else { if (((group_offset + 1) - (1 << (group_no - 1))) % 2 == 1) // Need to Insert 0 at middle op = 0; else // Need to Insert 1 at middle op = 1; aux_num = ((group_offset) - (1 << (group_no - 1))) / 2; } return constructNthNumber(group_no, aux_num, op); } // Driver code var n = 9; // Function Call console.log(getNthNumber(n)); // This code is contributed by Princi Singh ``` PHP ```<?php // Efficient PHP program to find n-th palindrome \$INT_SIZE=32; /* Construct the nth binary palindrome with the given group number, aux_number and operation type */ function constructNthNumber(\$group_no, \$aux_num,\$op) { global \$INT_SIZE; \$a=array_fill(0,\$INT_SIZE,0); \$num = 0; \$i = 0; \$len_f=0; // No need to insert any bit in the middle if (\$op == 2) { // Length of the final binary representation \$len_f = 2 * \$group_no; // Fill first and last bit as 1 \$a[\$len_f - 1] = \$a[0] = 1; /* Start filling the a[] from middle, with the aux_num binary representation */ while (\$aux_num) { // Get the auxiliary number's ith bit and // fill around middle \$a[\$group_no + i] = \$a[\$group_no - 1 - \$i] = \$aux_num & 1; \$aux_num = \$aux_num >> 1; \$i++; } } // Insert bit 0 in the middle else if (\$op == 0) { // Length of the final binary representation \$len_f = 2 * \$group_no + 1; // Fill first and last bit as 1 \$a[\$len_f - 1] = \$a[0] = 1; \$a[\$group_no] = 0; /* Start filling the a[] from middle, with the aux_num binary representation */ while (\$aux_num) { // Get the auxiliary number's ith bit and fill // around middle \$a[\$group_no + 1 + \$i] = \$a[\$group_no - 1 - \$i] = \$aux_num & 1; \$aux_num = \$aux_num >> 1; \$i++; } } else // Insert bit 1 in the middle { // Length of the final binary representation \$len_f = 2 * \$group_no + 1; // Fill first and last bit as 1 \$a[\$len_f - 1] = \$a[0] = 1; \$a[\$group_no] = 1; /* Start filling the a[] from middle, with the aux_num binary representation */ while (\$aux_num) { // Get the auxiliary number's ith bit and fill // around middle \$a[\$group_no + 1 + \$i] = \$a[\$group_no - 1 - \$i] = \$aux_num & 1; \$aux_num = \$aux_num >> 1; \$i++; } } /* Convert the number to decimal from binary */ for (\$i = 0; \$i < \$len_f; \$i++) \$num += (1 << \$i) * \$a[\$i]; return \$num; } /* Will return the nth binary palindrome number */ function getNthNumber(\$n) { \$group_no = 0; \$count_upto_group = 0; \$count_temp = 1; \$op=\$aux_num=0; /* Add number of elements in all the groups, until the group of the nth number is found */ while (\$count_temp < \$n) { \$group_no++; // Total number of elements until this group \$count_upto_group = \$count_temp; \$count_temp += 3 * (1 << (\$group_no - 1)); } // Element's offset position in the group \$group_offset = \$n - \$count_upto_group - 1; /* Finding which bit to be placed in the middle and finding the number, which we will fill from the middle in both directions */ if ((\$group_offset + 1) <= (1 << (\$group_no - 1))) { \$op = 2; // No need to put extra bit in middle // We need to fill this auxiliary number // in binary form the middle in both directions \$aux_num = \$group_offset; } else { if (((\$group_offset + 1) - (1 << (\$group_no - 1))) % 2) \$op = 0; // Need to Insert 0 at middle else \$op = 1; // Need to Insert 1 at middle \$aux_num = (int)(((\$group_offset) - (1 << (\$group_no - 1))) / 2); } return constructNthNumber(\$group_no, \$aux_num, \$op); } // Driver code \$n = 9; // Function Call print(getNthNumber(\$n)); // This code is contributed by mits ?> ```

Output
`27`

Time Complexity: O(1).
Auxiliary Space: O(1)

Reference:Â
https://www.codeproject.com/Articles/1162038/Finding-nth-Binary-Palindrome-in-Csharp
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