In the 6-stellar number pattern, it is observable that from 1 to 13, 12 is added; 13 to 37, 24 is added; 37 to 73, 36 is added; 73 to 121, 48 is added; 121 to 181, 60 is added.
From this information, it is possible to formulate an expression for the 6-stellar number at stage S7.
S7=1+12+2(12)+3(12)+4(12)+5(12)+6(12)
S7=1+12+24+36+48+60+72
S7=253
Moreover, using the same steps utilized for finding the general statement of the triangular pattern, a general statement for the 6-stellar number at stage Sn in terms of n can be fabricated.
Sn=1+12+2(12)+…+(n-2)12+(n-1)12+12n
+
Sn=12n+(n-1)12+(n-2)12+…+2(12)+12+1_______________________________________
2Sn=(1+12n)+(12+(n-1)12)+(2(12)+(n-2)12)+…+((n-2)12+2(12))+((n-1)12+12)+(12n+1)
2Sn=(1+12n)+(12+12n-12)+(24+12n-24)+…+(12n-24+24)+(12n-12+12)+(12n+1)
2Sn=(1+12n)+(12n)+(12n)+…+(12n)+(12n)+(12n+1)
2Sn=2+(12n)+(12n)+(12n)+…+(12n)+(12n)+(12n)
2Sn=2+(12n)(n-1)
Sn=1+6n(n-1)
To ensure the validity of this general statement, let us find the 6-stellar number at stage S3 using the general statement. According to the previous data, the 6-stellar number at stage S3 is 37.
Sn=1+6n(n-1)
S3=1+6(3)(3-1)
S3=1+6(3)(2)
S3=1+36
S3=37
The 6-stellar number at stage S3 acquired through the utilization of the general statement matches that of the pervious data, therefore, the general statement is genuine.
Now that we’ve identified the general statement for the 6-stellar number at stage Sn, let us consider, in a similar manner, stars with other values of p (different numbers of vertices).
5-Stellar Numbers
Firstly, let us examine stars with a p value of 5. In the following representation of a star with 5 vertices, the number of dots represents the 5-stellar number. Therefore, the first six 5-stellar numbers are as follows:
S1 S2 S3 S4
1 11 31 61
S5 S6
101 151
In the 5-stellar number pattern, it is clear that from 1 to 11, 10 is added; 11 to 31, 20 is added; 31 to 61, 30 is added; 61 to 101, 40 is added; 101 to 151, 50 is added.
Therefore, we can determine what the expression for the 5-stellar number at stage S7 would be.
S7=1+10+2(10)+3(10)+4(10)+5(10)+6(10)
S7=1+10+20+30+40+50+60
S7=211
Next, through steps similar to those for finding the general statement for the 6-stellar number at stage Sn, the general statement for the 5-stellar number at stage Sn, in terms of n, can be formed:
Sn=1+10+2(10)+…+(n-2)10+(n-1)10+10n
+
Sn=10n+(n-1)10+(n-2)10+…+2(10)+10+1_______________________________________
2Sn=(1+10n)+(10+(n-1)10)+(2(10)+(n-2)10)+…+((n-2)10+2(10))+((n-1)10+10)+(10n+1)
2Sn=(1+10n)+(10+10n-10)+(20+10n-20)+…+(10n-20+20)+(10n-10+10)+(10n+1)
2Sn=(1+10n)+(10n)+(10n)+…+(10n)+(10n)+(10n+1)
2Sn=2+(10n)+(10n)+(10n)+…+(10n)+(10n)+(10n)
2Sn=2+(10n)(n-1)
Sn=1+5n(n-1)
In order to be sure that the general statement is legit, let us find the 5-stellar number at stage S4 through the general statement. According to the gathered data from the diagrams of the stars with 5 vertices, the 5-stellar number at stage S4 should be 61.
Sn=1+5n(n-1)
S4=1+5(4)(4-1)
S4=1+5(4)(3)
S4=1+60
S4=61
The 5-stellar number at stage S4 received from the general statement matches that of the previous data. Hence, the general statement Sn=1+5n(n-1) is legit.
1-Stellar Numbers
Now, let us investigate stars with a p value of 1. Even though it will not be in a star-shape, investigating stars with a p value of 1 will be valuable in identifying the scope or limitations. As in the previous diagrams, in the following diagrams, the number of dots represents the 1-stellar number. The first six 1-stellar numbers are as follows:
S1 S2 S3 S4
1 4 10 19
S5 S6
31 46
In the 1-stellar number pattern, it is apparent that from 1 to 4, 3 is added; 4 to 10, 6 is added; 10 to 19, 9 is added; 19 to 31, 12 is added; 31 to 46, 15 is added.
From here, we can find the expression for the 1-stellar number at stage S7.
S7=1+3+2(3)+3(3)+4(3)+5(3)+6(3)
S7=1+3+6+9+12+15+18
S7=64
Next, the general statement for the 1-stellar number at stage Sn, in terms of n, is:
Sn=1+3+2(3)+…+(n-2)3+(n-1)3+3n
+
Sn=3n+(n-1)3+(n-2)3+…+2(3)+3+1_______________________________________
2Sn=(1+3n)+(3+(n-1)3)+(2(3)+(n-2)3)+…+((n-2)3+2(3))+((n-1)3+3)+(3n+1)
2Sn=(1+3n)+(3+3n-3)+(6+3n-6)+…+(3n-6+6)+(3n-3+3)+(3n+1)
2Sn=(1+3n)+(3n)+(3n)+…+(3n)+(3n)+(3n+1)
2Sn=2+(3n)+(3n)+(3n)+…+(3n)+(3n)+(3n)
2Sn=2+(3n)(n-1)
Sn=1+n(n-1)
Now, let’s test the validity of the general statement by substituting 6 for n. The answer we should get according to the diagrams is 46.
Sn=1+n(n-1)
S6=1+(6)(6-1)
S6=1+(6)(5)
S6=1+(30)
S6=1+45
S6=46
Since the 1-stellar number acquired from the general statement for stage S6 is the same as that from the diagrams, the general statement is valid.
2-Stellar Numbers
Finally, let us examine stars with a p value of 2. In the following diagrams, the number of dots represents the 2-stellar number. The first six 2-stellar numbers are as follows:
S1 S2 S3 S4
1 5 13 25
S5 S6
41 61
In the 2-stellar number pattern, it is evident that from 1 to 5, 4 is added; 5 to 13, 8 is added; 13 to 25, 12 is added; 25 to 41, 16 is added; 41 to 61, 20 is added.
Now, using the information above, the expression for the 2-stellar number at stage S7 can be identified.
S7=1+4+2(4)+3(4)+4(4)+5(4)+6(4)
S7=1+4+8+12+16+20+24
S7=85
Next, using the same steps as in the previous examples, the general statement, in terms of n, is:
Sn=1+3+2(4)+…+(n-2)4+(n-1)4+4n
+
Sn=4n+(n-1)4+(n-2)4+…+2(4)+4+1_______________________________________
2Sn=(1+4n)+(4+(n-1)4)+(2(4)+(n-2)4)+…+((n-2)4+2(4))+((n-1)4+4)+(4n+1)
2Sn=(1+4n)+(4+4n-4)+(8+4n-8)+…+(4n-8+8)+(4n-4+4)+(4n+1)
2Sn=(1+4n)+(4n)+(4n)+…+(4n)+(4n)+(4n+1)
2Sn=2+(4n)+(4n)+(4n)+…+(4n)+(4n)+(4n)
2Sn=2+(4n)(n-1)
Sn=1+2n(n-1)
At last, to test the legitimacy of the general statement, we will substitute 2 for n. The answer should match that of the diagram, which is 5.
Sn=1+2n(n-1)
S2=1+2(2)(2-1)
S2=1+2(2)(1)
S2=1+4
S2=5
As we had expected, the 2-stellar number from the general statement for stage S6 and that from the diagrams, are the same. Thus, the general statement for the 2-stellar number at stage Sn is correct.
p-Stellar Numbers
Hence, from the information gathered and the observations made, a general statement, in terms of p and n, that generates the sequence of p-stellar numbers for any value of p at stage Sn, can be produced.
From the data gathered, we can pull out the following:
General statement for 6-stellar numbers: Sn=1+6n(n-1)
General statement for 5-stellar numbers: Sn=1+5n(n-1)
General statement for 1-stellar numbers: Sn=1+n(n-1)
General statement for 2-stellar numbers: Sn=1+2n(n-1)
From this information, the general statement, in terms of p and n, that generates the sequence of p-stellar numbers for any value of p at stage Sn, is evidently:
Sn=1+pn(n-1)
The general statements of 6-stellar, 5-stellar, and 2-stellar numbers all correspond with the general statement that is in terms of p and n, generated above. The general statement for 1-stellar numbers will be discussed within scope or limitations, as it is the only one that does not correspond with the general statement for p-stellar numbers.
To ensure the validity of the general statement for the p-stellar number at stage Sn, let us conduct several tests. Note that (as stated at the beginning of the stellar numbers section) that p equals the number of vertices.
First, let us consider a star with 4 vertices. Using the general statement for the p-stellar number at stage Sn, let us find the stellar number at stage S5.
Sn=1+pn(n-1)
S5=1+4(5)(5-1)
S5=1+4(5)(4)
S5=1+80
S5=81
According to the general statement, the 4-stellar number at stage S5 is 81. Below is a diagram of a star with 4 vertices at stage S5:
S5
81
Consequently, the 4-stellar number (number of dots) revealed by the diagram matches that which was generated by the general statement for p-stellar numbers for any value of p at stage Sn. Thus far the general statement is valid.
Now, let us consider a star with 10 vertices. Utilizing our knowledge of the general statement for the p-stellar number at stage Sn, let us find the stellar number at stage S2.
Sn=1+pn(n-1)
S2=1+10(2)(2-1)
S2=1+10(2)(1)
S2=1+20
S2=21
According to the general statement, the 10-stellar number at stage S2 is 21. Below is a diagram of a star with 10 vertices at stage S2:
S2
21
The 10-stellar number (number of dots) disclosed by the diagram mirrors that which was generated by the general statement for p-stellar numbers for any value of p at stage Sn. Once again, the general statement hold true.
Regarding scope or limitations, the general statement holds true for any natural value of p except 1. The general statement for the 1-stellar number at stage Sn is Sn=1+n(n-1). Whereas, the general statement for the p-stellar number at stage Sn is Sn=1+pn(n-1). In order for the general statement to hold true for p=1, the in the 1-stellar number general statement at stage Sn would have to be 1. However, in that case, the general statement for the 1-stellar number at stage Sn wouldn’t satisfy reality, as you would receive incorrect 1-stellar numbers from the general statement. Other than 1, p can be equal to any natural (i.e. 2, 3, 4, 5, 6, etc). Furthermore, n can also equal any natural number, including 1.
Hence,
p , p1
n
Furthermore, the result of the general statement could never be negative as a figure can’t have negative vertices or stages. In addition, the result of the general statement will continuously near infinity as p and n are increased.
Arriving at the general statement was mostly based on the previously gathered information. Firstly, I pulled out the general statements of the 6-stellar, 5-stellar, 1-stellar, and 2-stellar numbers. Those general statements were:
6-stellar numbers: Sn=1+6n(n-1)
5-stellar numbers: Sn=1+5n(n-1)
1-stellar numbers: Sn=1+n(n-1)
2-stellar numbers: Sn=1+2n(n-1)
I noticed that in general statement for the 6-stellar number at stage Sn included the number 6 in it:
1+6n(n-1)
Next I noticed that the general statement for the 5-stellar number at stage Sn included the number 5 in it:
1+5n(n-1)
Then, I perceived that the general statement for the 2-stellar number at stage Sn included the number 2 in it:
1+2n(n-1)
The reason why the same pattern didn’t exist for the 1-stellar numbers was discussed in scope and limitations. Therefore, from the observed pattern, I took into account that those numbers that matched represented the number of vertices, which p also represented. Hence, I formed my general statement for the p-stellar numbers for any value of p at stage Sn, which was:
Sn=1+pn(n-1)
