We can calculate the term by adding to the previous number of dots. For example, the value for equals to 15 +6 which is 21. The same applies for the other values of .
The next three terms are represented by this diagram:
When we want to find the expression for this sequence, we have to know which difference is constant.
Seq.: 1, 3, 6, 10, 15…
1st diff.: 2 3 4 5…
2nd diff.: 1 1 1…
We found from the above that the second difference is constant. Therefore the expression will be . We can know the coefficient of the by dividing the 2nd difference by 2 and the result is the coefficient of .
Therefore the first part of the expression is
To know the rest, we must list the sequence and subtract it from the original sequence.
The result from subtraction is a linear sequence (). We can work out the expression of this sequence and add it to to get the expression for triangular numbers.
Sequence: 0.5, 1, 1.5, 2, 2.5, 3, 3.5…
1st diff.: 0.5 0.5 0.5 0.5 0.5 0.5
2nd diff.: 0 0 0 0 0
If the general formula of linear sequences is , then will be 0.5 and will be 0.
The expression of this sequence will be
Therefore the expression of triangular numbers will be:
We can check this by trying the expression with the 9th term..The difference between 45 and 36 is 9, therefore the expression is correct because this difference fits with the previous ones.
The graph below shows that the expression is correct.
Stellar numbers
The next type of numbers we are considering is stellar numbers. Stellar numbers are sequences of dots. The 1st term is 1 dot. After that, this dot is surrounded by a star in the 2nd term. The previous star is surrounded by a larger star in the following terms. The value of is the number of vertices the star has and this is also called -stellar number.
6-stellar numbers
These are stellar numbers of stars with 6 vertices.
To know the expression of this sequence, we must know which difference is constant.
Sequence: 1, 13, 37, 73, 121, 181
1st diff.: 12 24 36 48 60
2nd diff.: 12 12 12 12
This is a quadratic sequence () because the 2nd difference is constant.is equal to half the second difference which is 6.
The first part of the expression is
We have to subtract the sequence from the original sequence to get the linear part of this expression (i.e. ).
The expression of this sequence will be the final part of the original sequence.
Sequence: -5, -11, -17, -23, -29, -35…
1st diff.: -6, -6, -6, -6, -6
The expression is now
However, it is not complete. If we try it on S2, it equals 12 not 13. So we will add 1 as c to the expression.
So the expression of S7 will be
We know that this is true because the difference between 253 and 181 is 72 which fit with the 1st differences. We can also count the stars in the following diagram.
The following graph shows the accuracy of the expression
5 Stellar numbers
These are stellar numbers of stars with 5 vertices.
S1 S2 S3 S4 S5 S6
SN: 1, 11, 31, 61, 101, 151…
Each star consists of a pentagon with a triangle attached to it. The number of dots in all the pentagons is 5. The number of dots in each triangle is 2n-3. I only calculated the last shell and added it to the previous number of dots to get the stellar number.
Seq.: 1, 11, 31, 61, 101, 151…
1st Diff.: 10 20 30 40 50
2nd Diff.: 10 10 10 10
This sequence is quadratic (). The first part of the expression is . To know the rest of the expression, we will need to subtract the sequence from the original sequence.
We now have a linear sequence ( ). This expression will be added to the first part of the stellar number sequence.
Seq.: -4, -9, -14, -19, -24, -29…
1st Diff.: -5 -5 -5 -5 -5
If we try the expression now, it will not work. We have to add +1 to it.
The expression for S7 is
We can check that this is true by subtracting 151 from 211 and the result should fit in the 1st differences row. The difference is 60 and fits in the 1st differences row. We can also count the stars in the diagram.
General Statement:
The functions of these two stellar numbers are very similar as in this graph.
After knowing the expressions for stellar numbers of 2 different vertices, we found that the common units in the expression are
And we found that the coefficient of and is the number of vertices that the star has.
The general statement for every star with p vertex is:
Where is greater than 2 and is greater than or equal to 1.
This statement works only for real numbers and positive numbers because you can’t have a negative star or half a star.
