
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
March 5th, 2011, 10:11 AM  #1 
Joined: Jun 2010 Posts: 38 Thanks: 0  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. Thanks 
March 5th, 2011, 10:36 AM  #2 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,808 Thanks: 268 Math Focus: The calculus  Re: Stellar Numbers
It seems your numbers follow the pattern: where is the nth triangular number, giving: 
March 5th, 2011, 10:45 AM  #3 
Joined: Jun 2010 Posts: 38 Thanks: 0  Re: Stellar Numbers
Can I ask why did you use the triangular numbers formula?

March 5th, 2011, 10:51 AM  #4 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,808 Thanks: 268 Math Focus: The calculus  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. 
March 5th, 2011, 10:54 AM  #5  
Joined: Jun 2010 Posts: 38 Thanks: 0  Re: Stellar Numbers Quote:
 
March 5th, 2011, 11:35 AM  #6 
Joined: Jun 2010 Posts: 38 Thanks: 0  Re: Stellar Numbers
One more question is that why did you use (n1) What was the reasoning behind it? 
March 5th, 2011, 11:46 AM  #7 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,808 Thanks: 268 Math Focus: The calculus  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 
March 5th, 2011, 01:55 PM  #8 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,808 Thanks: 268 Math Focus: The calculus  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: 
April 15th, 2011, 07:38 AM  #9 
Joined: Apr 2011 Posts: 1 Thanks: 0  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! =) 
April 15th, 2011, 12:18 PM  #10 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,808 Thanks: 268 Math Focus: The calculus  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.


Tags 
numbers, stellar 
Search tags for this page 
stellar numbers triangular numbers general formula,stellar numbers theory,the formula of stellar numbers,6 stellar numbers formula,what are stellar numbers
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Can complex numbers (and properties of complex numbers) be..  jonas  Complex Analysis  2  October 13th, 2014 03:03 PM 
The paradox between prime numbers and natural numbers.  Eureka  Number Theory  4  November 3rd, 2012 03:51 AM 
Expectation of picking all N numbers from N distinct numbers  abcbc  Advanced Statistics  1  October 1st, 2012 04:14 PM 
Numbers that are not complex numbers  Mathforum1000  Number Theory  2  June 20th, 2012 04:18 PM 
perfect numbers and amicable numbers  soandos  Number Theory  2  November 27th, 2007 05:42 AM 