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, ….
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.
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)
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