A number is termed as star number, if it is a centered figurate number that represents a centered hexagram (six-pointed star) similar to chinese checker game. The few star numbers are 1, 13, 37, 73, 121, 181, 253, 337, 433, ….
Examples:
Input : n = 2 Output : 13 Input : n = 4 Output : 73 Input : n = 6 Output : 181
If we take few examples, we can notice that the n-th star number is given by the formula:
n-th star number = 6n(n - 1) + 1
Below is the implementation of above formula.
C++
// C++ program to find star number #include <bits/stdc++.h> using namespace std; // Returns n-th star number int findStarNum(int n) { return (6 * n * (n - 1) + 1); } // Driver code int main() { int n = 3; cout << findStarNum(n); return 0; } |
Java
// Java program to find star number import java.io.*; class GFG { // Returns n-th star number static int findStarNum(int n) { return (6 * n * (n - 1) + 1); } // Driver code public static void main(String args[]) { int n = 3; System.out.println(findStarNum(n)); } } // This code is contributed // by Nikita Tiwari. |
Python3
# Python3 program to # find star number # Returns n-th # star number def findStarNum(n): return (6 * n * (n - 1) + 1) # Driver code n = 3print(findStarNum(n)) # This code is contributed by Smitha Dinesh Semwal |
C#
// C# program to find star number using System; class GFG { // Returns n-th star number static int findStarNum(int n) { return (6 * n * (n - 1) + 1); } // Driver code public static void Main() { int n = 3; Console.Write(findStarNum(n)); } } // This code is contributed // by vt_m. |
PHP
<?php //PHP program to find star number // Returns n-th star number function findStarNum($n) { return (6 * $n * ($n - 1) + 1); } // Driver code $n = 3; echo findStarNum($n); // This code is contributed by ajit ?> |
Javascript
<script> // Javascript program to find star number // Returns n-th star number function findStarNum(n) { return (6 * n * (n - 1) + 1); } // Driver code let n = 3; document.write(findStarNum(n)); // This code is contributed by rishavmahato348. </script> |
Output :
37
Time complexity: O(1) since performing constant operations
Space complexity: O(1) since using constant variables
Interesting Properties of Start Numbers:
- The digital root of a star number is always 1 or 4, and progresses in the sequence 1, 4, 1.
- The last two digits of a star number in base 10 are always 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93.
- The generating function for the star numbers is
x*(x^2 + 10*x + 1) / (1-x)^3 = x + 13*x^2 + 37*x^3 +73*x^4 .......
- The star numbers satisfy the linear recurrence equation
S(n) = S(n-1) + 12(n-1)
References :
http://mathworld.wolfram.com/StarNumber.html
https://en.wikipedia.org/wiki/Star_number
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