The sequence Sn in this case is:

(1,13,37,73,121,181…)

The method that was used for the table heading in cellA1 and B1. And made an excel formula in A2 namely A1-B1. So the difference was obtained. For the graph A1 was plotted in relation to B1.

One can observe from the graph and the table that since difference of Sn values seem to be an arithmetic sequence then Sn is likely to be generated by a quadratic formula. Whith this knowledge following process is obtained.

Observing Sn one realizes that every value of Sn can be expressed as the difference of the difference times a number plus 1. So,

Sn=

From the table it can be assumed that is 12 consistently; a value for can be inserted,

Sn=12

Now x is investigated, one observes that the values of x are of some kind of number sequence which will be called Cn. Cn will be investigated by exemplifying with the value of S7.

S7 = 253

S7= 252+1

S7= 12 21 +1

In order to receive a general statement this expression will be wrote in terms of n.

Sn = 12 Cn+1

Calculating the values for Cn following values are obtained.

(0,1,3,6,10,15…)

Looking at this sequence one notices that it seems to be similar to the triangular sequence derived previously. In fact, observing and comparing the sequences one can see that the Cn corresponds to T(n-1).

Now I can substitute Cn with T(n-1)

Thus,

Sn= 12 T(n-1)+1

Since,

Tn =

Sn will be,

Looking at the general statement of the star it can be noticed that the coefficient 6 is equal to the number of vertices in the stellar shape. Assuming 6 represents p, 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 derived. Expressing this mathematically,

Sn=pn(n-1)+1

Testing this hypothesis requires an investigation of the vertices-value p. First, a star with five vertices is investigated.

When p =5 the star has a five stellar number with the following values for Sn.

S1=1

S2=11

S3=31

S4=61

From the table one can see that for the previously derived statement,,.

Investigating x one can again observe that it is again a sequence Cn (0,3,6,10,15…). Thus x=Tn-1 again.

Investigating this for the value S7,

S7 = 211

S7= 210+1

S7= 57(7-1) +1

To find the general statement for the 5-steller number, 7 is replaced by n.

Sn=5n(n-1)+1

Second, a star with seven vertices is investigated. When p =7 the star has a seven stellar number with the following values for Sn.

S1= 1

S2= 15

S3= 43

S4= 85

From the table one can see that for the previously derived statement,,.

Investigating x one can again observe that it is again a sequence Cn (0,3,6,10,15…). Thus x=Tn-1 once again.

Investigating this for the value S7,

S7 = 295

S7= 294+1

S7= 77(7-1) +1

To find the general statement for the 7-stellar number, 7 is replaced by n.

Sn=7n(n-1)+1

To find the general statement for the 7-stellar numbers, 7 is replaced by n.

Sn=7n(n-1)+1

Testing other values for p one can see that tthey seem to follow the pattern.

*The values were calculated using a TI 84 calculator.

Observing this table it seems like the hypothesis Sn=pn(n-1)+1 is appliable.

At this point one could test the formula in many new situations, however, taking a more analytical approach will perhaps help to prove this general statement.

Investigating the 6-stellar number it was found that the stars follow the sequence Sn at the initial condition that .It was observed that Sn could be written in terms of Cn( another sequence) times the coefficient +1. Since was 12 for all investigated values, it was found that the sequence Cn is equal to T(n-1); having derived Tn previously, Cn could be substituted in terms of n. Thus, we could derive that Sn= 6n(n-1)+1.

At this point it was noticed that 6 is the number of vertices in the Star so we investigated the Hypothesis Sn=pn(n-1)+1.After investigating the values 3-10 for p it seems like the general statement is valid . Thus, the formula seems to work for all positive integers. However, when p=2 the question arouses whether a shape with 2 vertices is a stellar shape? Investigating this one finds that the values obtained for Sn when p=2 are (1,5,13,25,41,61…). In fact, one can say that the shape could actually be a Diamond. Ever since the triangular sequence is included in the stellar sequence as one may argue that a stellar shape with 2 vertices can be existent, but this remains an open question. Another value of p that aroused my attention is 1 one finds that the values obtained for Sn when p=1 are (1,3,6,10,15…) in fact one observes that this sequence is equal to Tn. Which would suggest that a stellar shape with the value p=1 is a triangle, one could at this point assume that a stellar shape with p=1 is a triangle. However, whether a triangle can be defined as a stellar shape with 1 vertex remains an open question.

Furthermore, investigating negative values, one can observe that for the general statement Sn=pn(n-1)+1

If p=negativeSn=negative

If n=negativeSn=negative

If p=negative and n=negativeSn=positive

It seems that theoretically any rational number can be inserted for n and p in this formula; however, the shape then is questionable. If a number of negative dots and vertices can form a geometrical shape is a complex issue, which can not be investigated at this point. Also, the shapes when p= 2 and p= 1 if these vertices lead to actual Stellar shapes is open, but it seems likely since the sequence of Tn is included in the sequence Sn.

Thus, the conclusion can be drawn that when the conditions,

n>0

p>2

therefore if p is a fixed positive integer,

(not that infinite is written out instead of ∞, this was due to technical problems)

The formula Sn=pn(n-1)+1 can be applied to derive a certain Stellar shape.

When these conditions do not apply a clear pattern of a sequence can still be observed; however, if all the values lead to definite Stellar shapes remains an open question.