Generate sequence with equal sum of adjacent integers

Last Updated : 19 Sep, 2023

Given an integer N, find a sequence of N integers such that the sum of all integers in the sequence is equal to the sum of any two adjacent integers in the sequence.

Examples:

Input: N = 4
Output: 1 -1 1 -1
Explanation: Total sum of the four integers is 0 and the sum of any two adjacent integers is also 0.

Input: N = 5
Output: 1 -2 1 -2 1
Explanation: Total sum of the four integers is -1 and the sum of any two adjacent integers is also -1.

Approach: To solve the problem follow the below idea:

For even n, we can use the array [-1, 1, -1, 1, ..., -1, 1] as a solution. This array has a length of n, and the sum of any two adjacent elements is 0, as well as the sum of the whole array.
For odd n, we can construct an array s of length n using two fixed values a and b, such that s_i-1= s_i+1 for i = 2, 3, ...n-1. Specifically, we can set s₁ = a and s2 = b, and then create the array s = [a, b, a, b, ..., a, b, a]. We can then solve for 'a' and 'b' using the fact that the sum of any two adjacent elements and the sum of the whole array must be equal.
If we let k be a positive integer such that n = 2k+1, then we can find values for 'a' and 'b' that satisfy the conditions. Specifically, we can set a = k-1 and b = -k, which produces the array [k-1, -k, k-1, -k, ..., k-1, -k, k-1]. This array has a length of n, and the sum of any two adjacent elements is k-1-k = -1, as well as the sum of the whole array being k(k-1)-(k-1)k = 0.
However, there is no solution for n = 3, as a = 0 and b = 0 do not satisfy the conditions. In general, the array we constructed will not work if a or b is equal to 0. Therefore, we must have k ≥ 2 to ensure that both a and b are nonzero.
Overall, this approach shows that if there is a solution for even n, then there is a solution for odd n greater than or equal to 5.

Follow the steps to solve the problem:

Initialize an integer variable 'num' with the value N/2.
If N is even print 1, -1, ........., 1, -1
If N is odd :
- iterate through the sequence of integers from 1 to N.
- If i is odd, print -num, otherwise print num-1.

Below is the implementation for the above approach:

C++

// C++ code for the above approach: #include <bits/stdc++.h> using namespace std;  // Function to solve the problem void solve(int n) {      // If n is odd     if (n % 2) {          // Calculate the number to be         // added and subtracted         // alternatively         int num = n / 2;          // Iterate through the sequence         // and print the values         for (int i = 0; i < n; i++) {             if (i % 2)                  // If i is odd, print                 // the negative number                 cout << -num << " ";              // Otherwise, print the             // positive number             else                 cout << num - 1 << " ";         }          // Print a newline character         // at the end         cout << endl;     }      // If n is even     else {          // Iterate through the sequence         // and print the values         for (int i = 0; i < n; i++) {              // If i is odd, print -1             if (i % 2)                 cout << -1 << " ";              // Otherwise, print 1             else                 cout << 1 << " ";         }          // Print a newline character         // at the end         cout << endl;     } }  // Driver function int main() {     int n = 9;      // Call the solve function with n = 9     solve(9); }

Java

// JAVA code for the above approach:  public class GFG {     // Function to solve the problem     public static void solve(int n) {         // If n is odd         if (n % 2 == 1) {             // Calculate the number to be added and             // subtracted alternatively             int num = n / 2;              // Iterate through the sequence and print the values             for (int i = 0; i < n; i++) {                 if (i % 2 == 1)                     // If i is odd, print the negative number                     System.out.print(-num + " ");                 else                     // Otherwise, print the positive number                     System.out.print(num - 1 + " ");             }             // Print a newline character at the end             System.out.println();         }         // If n is even         else {             // Iterate through the sequence and print the values             for (int i = 0; i < n; i++) {                 // If i is odd, print -1                 if (i % 2 == 1)                     System.out.print(-1 + " ");                 // Otherwise, print 1                 else                     System.out.print(1 + " ");             }             // Print a newline character at the end             System.out.println();         }     }      // Driver function     public static void main(String[] args) {         int n = 9;          // Call the solve function with n = 9         solve(9);     } }   // this code is contributed by rambabuguphka

Python3

def solve(n):     # If n is odd     if n % 2:         # Calculate the number to be added and subtracted alternatively         num = n // 2          # Iterate through the sequence and print the values         for i in range(n):             if i % 2:                 # If i is odd, print the negative number                 print(-num, end=" ")             else:                 # Otherwise, print the positive number                 print(num - 1, end=" ")          # Print a newline character at the end         print()      # If n is even     else:         # Iterate through the sequence and print the values         for i in range(n):             # If i is odd, print -1             if i % 2:                 print(-1, end=" ")             # Otherwise, print 1             else:                 print(1, end=" ")          # Print a newline character at the end         print()   # Driver function if __name__ == "__main__":     n = 9      # Call the solve function with n = 9     solve(9)  # This code is contributed by shivamgupta310570

using System;  public class GFG {     // Function to solve the problem     static void Solve(int n)     {         // If n is odd         if (n % 2 != 0)         {             // Calculate the number to be             // added and subtracted             // alternatively             int num = n / 2;              // Iterate through the sequence             // and print the values             for (int i = 0; i < n; i++)             {                 if (i % 2 != 0)                      // If i is odd, print                     // the negative number                     Console.Write(-num + " ");                  // Otherwise, print the                 // positive number                 else                     Console.Write(num - 1 + " ");             }              // Print a newline character             // at the end             Console.WriteLine();         }          // If n is even         else         {             // Iterate through the sequence             // and print the values             for (int i = 0; i < n; i++)             {                 // If i is odd, print -1                 if (i % 2 != 0)                     Console.Write(-1 + " ");                  // Otherwise, print 1                 else                     Console.Write(1 + " ");             }              // Print a newline character             // at the end             Console.WriteLine();         }     }      // Driver function     public static void Main(string[] args)     {         int n = 9;          // Call the solve function with n = 9         Solve(n);     } }  // This code is contributed by akshitaguprzj3

JavaScript

// Function to solve the problem function solve(n) {      // If n is odd     if (n % 2) {              // Calculate the number to be         // added and subtracted         // alternatively         let num = Math.floor(n / 2);                  // Iterate through the sequence         // and print the values         for (let i = 0; i < n; i++) {             if (i % 2)                 // If i is odd, print                 // the negative number                 console.log(-num + " ");                          // Otherwise, print the             // positive number             else                 console.log(num - 1 + " ");         }                  // Print a newline character         // at the end         console.log();     }          // If n is even     else {              // Iterate through the sequence         // and print the values         for (let i = 0; i < n; i++) {                      // If i is odd, print -1             if (i % 2)                 console.log(-1 + " ");                              // Otherwise, print 1             else                 console.log(1 + " ");         }                  // Print a newline character         // at the end         console.log();     } }  // Driver function let n = 9; solve(9);

Output

3 -4 3 -4 3 -4 3 -4 3

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

Generate minimum sum sequence of integers with even elements greater

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Generate sequence with equal sum of adjacent integers

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