Program to print pentatope numbers upto Nth term in C

A pentatope number is a figurate number that represents the number of points in a pentatope (4-dimensional simplex). These numbers form the fifth diagonal of Pascal's triangle. The nth pentatope number represents the number of ways to choose 4 items from n+3 items.

Syntax

pentatope(n) = n * (n + 1) * (n + 2) * (n + 3) / 24

Mathematical Formula

The formula for the nth pentatope number is −

pentatope(n) = C(n+3, 4) = n * (n + 1) * (n + 2) * (n + 3) / 24

The first few pentatope numbers are −

1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365...

Example: Calculate Nth Pentatope Number

This example calculates and displays the nth pentatope number using the mathematical formula −

#include <stdio.h>

int calculatePentatope(int n) {
    return n * (n + 1) * (n + 2) * (n + 3) / 24;
}

int main() {
    int n;
    
    printf("Enter the term number: ");
    scanf("%d", &n);
    
    int result = calculatePentatope(n);
    printf("The %dth pentatope number is: %d<br>", n, result);
    
    return 0;
}

Enter the term number: 5
The 5th pentatope number is: 70

Example: Print Pentatope Numbers up to Nth Term

This example prints all pentatope numbers from the 1st term up to the nth term −

#include <stdio.h>

int calculatePentatope(int n) {
    return n * (n + 1) * (n + 2) * (n + 3) / 24;
}

int main() {
    int n;
    
    printf("Enter the number of terms: ");
    scanf("%d", &n);
    
    printf("\nPentatope numbers up to %d terms:<br>", n);
    
    for (int i = 1; i <= n; i++) {
        printf("%d ", calculatePentatope(i));
    }
    printf("<br>");
    
    return 0;
}

Enter the number of terms: 8
Pentatope numbers up to 8 terms:
1 5 15 35 70 126 210 330

Key Points

• Pentatope numbers are the 4th tetrahedral numbers or 5th diagonal of Pascal's triangle.
• The formula involves combinations: C(n+3, 4) where n is the term number.
• Each pentatope number represents points in a 4-dimensional simplex.
• The sequence starts from n=1, giving the first pentatope number as 1.

Conclusion

Pentatope numbers are calculated using the combination formula C(n+3, 4). The simple multiplication and division approach makes it efficient to generate these figurate numbers for mathematical applications.