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March 5th, 2011, 10:11 AM   #1
 
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Stellar Numbers

The first four stages for a star with 6 vertices have 1,13,37,73 number of dots. I have found a pattern that if I subtract one from each term it's a multiple of 12. So I need to figure out an equation for it.

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March 5th, 2011, 10:36 AM   #2
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Re: Stellar Numbers

It seems your numbers follow the pattern:

where is the nth triangular number, giving:

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March 5th, 2011, 10:45 AM   #3
 
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Re: Stellar Numbers

Can I ask why did you use the triangular numbers formula?
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March 5th, 2011, 10:51 AM   #4
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Re: Stellar Numbers

You observed that each number was some multiple of 12 plus 1, so I used this to write the numbers as:

12(0) + 1, 12(1) + 1, 12(3) + 1, 12(6) + 1

and I recognized the sequence 0, 1, 3, 6 as being the triangular numbers.
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March 5th, 2011, 10:54 AM   #5
 
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Re: Stellar Numbers

Quote:
Originally Posted by MarkFL
You observed that each number was some multiple of 12 plus 1, so I used this to write the numbers as:

12(0) + 1, 12(1) + 1, 12(3) + 1, 12(6) + 1

and I recognized the sequence 0, 1, 3, 6 as being the triangular numbers.
Wow you are a math god
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March 5th, 2011, 11:35 AM   #6
 
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Re: Stellar Numbers

One more question is that why did you use (n-1)
What was the reasoning behind it?
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March 5th, 2011, 11:46 AM   #7
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Re: Stellar Numbers

Because the first stellar number was a function of the zeroth triangular number, and the second stellar number was a function of the first triangular number, etc. We could say 1 is the zeroth stellar number and 13 is the first, etc. Then we would have:





It just depends on how you define it. We could also define the stellar numbers recursively:

where
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March 5th, 2011, 01:55 PM   #8
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Re: Stellar Numbers

Armed only with the recursive formula, we could find the closed form as follows:

We need to transform the relation into a linear homogeneous relation with constant coefficients. So we may write:

(1)

(2)

Subtract (2) from (1) to obtain:



(3)

(4)

Subtract (4) from (3) to obtain:

which we may write as:

(5)

Now, we assume this has a solution of the form and substituting this into (5) we obtain:



Dividing through by we obtain the characteristic equation:

so we have the general solution:



Now we use initial conditions to find the constants:







From the first equation we have so the second equation becomes:

which we may substitute into the third equation:



thus we have:

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April 15th, 2011, 07:38 AM   #9
 
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Re: Stellar Numbers

To MarkFL:

Hi, I saw the work that you did on Stellar Numbers on this forum and I had a question about them and you seem to be very knowledgeable in this subject! As part of a 'math essay', I am required to write an introductory paragraph to Stellar Numbers and I was just wondering if you happened to know what real life applications triangular/stellar numbers have?

Itd be awesome if you could somehow help me with that, thanks! =)
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April 15th, 2011, 12:18 PM   #10
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Re: Stellar Numbers

To be completely honest, I had never heard of stellar numbers before this topic. I just happened to recognize that the stellar numbers appeared to be a function of triangular numbers, which come from the summation of consecutive natural numbers beginning with 1.
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