Print n terms of Newman-Conway Sequence

The Newman-Conway Sequence is a fascinating mathematical sequence that generates integers using a recursive formula. It starts with 1, 1 and each subsequent term is calculated based on previous values in the sequence.

Syntax

P(n) = P(P(n - 1)) + P(n - P(n - 1))
where P(1) = P(2) = 1

Algorithm

The algorithm to generate n terms of Newman-Conway sequence is −

START
Step 1 ? Input variable n (e.g. 20)
Step 2 ? Initialize variables as i, p[n+1], p[1]=1, p[2]=1
Step 3 ? Loop For i=3 and i<=n and i++
   Set p[i] = p[p[i - 1]] + p[i - p[i - 1]]
   Print p[i]
Step 4 ? End Loop For
STOP

Example

Here's a complete C program to print the first n terms of Newman-Conway sequence −

#include <stdio.h>

int main() {
    int n = 20, i;
    int p[n + 1];
    
    p[1] = 1;
    p[2] = 1;
    
    printf("Newman-Conway Sequence is: ");
    printf("%d %d ", p[1], p[2]);
    
    for (i = 3; i <= n; i++) {
        p[i] = p[p[i - 1]] + p[i - p[i - 1]];
        printf("%d ", p[i]);
    }
    printf("<br>");
    
    return 0;
}

Newman-Conway Sequence is: 1 1 2 2 3 4 4 4 5 6 7 7 8 8 8 8 9 10 11 12

How It Works

The sequence works by using the recursive formula where each term depends on two previous terms. For example:

• P(3) = P(P(2)) + P(3-P(2)) = P(1) + P(2) = 1 + 1 = 2
• P(4) = P(P(3)) + P(4-P(3)) = P(2) + P(2) = 1 + 1 = 2
• P(5) = P(P(4)) + P(5-P(4)) = P(2) + P(3) = 1 + 2 = 3

Conclusion

The Newman-Conway sequence demonstrates how complex patterns can emerge from simple recursive formulas. This implementation efficiently computes the sequence using dynamic programming with O(n) time and space complexity.