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# Stellar Numbers (Part 2)

The next part of our task is to consider **stellar **(star) shapes with *p *vertices, leading to *p*-stellar numbers. The first four representations for a star with six vertices are shown in the four stages S1–S4 below. The 6-stellar number at each stage is the total number of dots in the diagram.

Again in this task like in previous one I will try to present this diagram in form of table because of easier work. I simply counted every dot in each star and put into the table.

Since each number represent sum, we should find actual numbers in this sequence and from general formula for sum which says that: * S_{n}=U_{1}+U_{2} + U_{3 }+ U_{4 }+ … + U_{n }*we can calculate all numbers in this sequence.

So; if * S_{n}=U_{1}+U_{2} + U_{3 }+ U_{4 }+ … + U_{n }*then

**and since we have**

*S*_{2}=U_{2}+U_{1 }**S**which is

_{2}**13**and we know

**U**which is

_{1 }**1**we can calculate

**, we can therefore say that:**

*U*_{2}; U_{2}= S_{2}-U_{1}**so we can calculate this and**

*U*_{2}=13-1

*U*_{2}=12.So; if ** S_{n}=U_{1}+U_{2} + U_{3}+U_{4}+…+U_{n }**then

**and since**

*S*_{3}=U_{1}+U_{2}+U_{3 }**then**

*U*_{1}+U_{2}=S_{2 }*so*

**S**,_{3}=S_{2}+U_{3}**and from this**

*U*_{3}=S_{3}-S_{2 }**we can calculate that**

*U*_{3}=37-13

*U*_{3}=24*.*So; if ** S_{n}=U_{1}+U_{2} + U_{3}+U_{4}+…+U_{n }**then

**and since**

*S*_{4}=U_{1}+U_{2}+U_{3}+U_{4 }**then**

*U*_{1}+U_{2}+U_{3}=S_{3 }**so**

*S*_{4}=S_{3}+U_{4},**and from this**

*U*_{4}=S_{4}-S_{3 }*we can calculate that*

**U**_{4}=73-37

*U*_{4}=36.When we know this information then we can calculate the difference and check if the difference is the same each time. That can be easily checked by using this formula: ** d=U_{n}-U_{n-1. }**This time I will calculate like this:

**and if this is true then the difference is same for these two terms in this sequence.**

*U*_{3}-U_{2}= U_{3}-U_{4 }So, by using ** U_{3}-U_{2} = U_{4}-U_{3 }**calculations go like this:

**so the difference in both cases is twelve and because of this I will take this sequence**

*U*_{3}-U_{2}= U_{4}-U_{3}=> 24-12 = 36-24 => 12 = 12__partially__arithmetic. I take it partially arithmetic because difference between second term and the first one isn’t

**12**but because of the fact that after first term which is one in this case (that represents 1 dot) stars won’t get any additional points like this one in the middle I can work with other terms and take that part (after first term 1) of sequence as arithmetic.

Since we know what is the common difference in next terms we can calculate them in very simple way, by using this formula: *U _{n}=U_{n-1}+d*

So, since we need to calculate number of dots in each stage up to stage** S _{6}** I will calculate

**and**

*U*_{5 }**because I already have**

*U*_{6 }**Using this simple formula I will calculate**

*U*_{1to4. }**So,**

*U*_{5. }

*U*_{5}*=**U*_{4}*+d =>**U*_{5 }*= 36+12 =>**U*_{5 }*= 48.*Before I calculate * U_{6 }*I can calculate how much dots is there in

*That can be calculated by adding the*

**S**_{6. }*S*stage with number of dots in the next layer of star which is

_{4 }

**U**_{5. }So,**S**_{5}= S_{4}+U_{5 }=> S_{5 }= 73 + 48 => S_{5 }=121**.**This number

**shows how many dots is there in stage**

*S*_{5 }**of the diagram.**

*S*_{5 }Next thing I need to do is to calculate ** S_{6 }**stage of the diagram and I should do that by calculating

**and then adding that number to the number of dots in stage**

*U*_{6 }**and by introducing our numbers into this formula we get**

*S*So,_{5- }*U*=_{6 }*U*+_{5 }*d**Now I need to calculate how many dots there is in stage*

**U**._{6 }= 48 + 12 => U_{6 }= 60**of the diagram and that can be calculated by adding**

*S*_{6 }*to the*

**U**_{6 }*Hence*

**S**_{5. }**and**

*U*= 60_{6 }**we can calculate that**

*S*= 121_{5 }

*S*_{6 }= 121 + 60 => S_{6 }= 181.Now when I know how many dots are in new layer of star and how many dots are there in each stage I can easily draw those stars. They look like this:

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