Stellar Numbers

Triangular Numbers:

Diagram 1 : Triangular Numbers

T1 = 1                T2 = 3                T3 = 6                T4 = 10                T5 = 15                T6 = 21        

T7 = 28                T8 = 36

First Arithmetic Progression (AP) = T2 –T1

                                 = 3 – 1

                                 = 2

Difference from Arithmetic Progression (d) = (T3 – T2) – (T2 –T1)

                                             = (6 – 3) – (3 – 1)

                                             = 3 – 2

                                                 = 1

To determine the number of dots in one triangle there are two possible formulas which are possible. The first is that where Tn is equal to that of the first triangle, with the addition of the Sum of n -1. This formula can be simplified after substituting the Arithmetic Progression and difference into the Sum of n – 1. This gives the final formula in which Tn equals one (1) and n-1 divided by two (2) multiplied by four (4) and n minus two (2).

The other formula is that of when Tn is equal to n plus n-1 divided by two (2) multiplied by, two AP plus d of n take two (2). This formula can also be simplified and this gives the result of Tn equal to that of n divided by two, multiplied by that of n with the addition of one (1).

Tn = T1 + (Sumn – 1)

Tn = T1 + [(n-1) ÷2] x [2 x AP + (n – 1 -1) d]

Tn = 1 + [(n-1) ÷2] x [2 x 2 + (n – 2)]

Tn = 1 + [(n-1) ÷2] x [4 + (n – 2)]

Or

Tn = n + [(n-1) ÷2] x [2 x T1 + (n – 1 -1) d]

Tn = n + [(n-1) ÷2] x [2 x 1 + (n – 1 -1)1]

Tn = n + [(n-1) ÷2] x [2 + (n – 2)]

Tn = (2n + n2 –n) ÷ 2

Tn = (n + n2) ÷ 2

Tn = (n ÷ 2) (n + 1)

Table 1: Triangular Numbers

The Arithmetic Progression for the 6-point stellar star is that of twelve (12), or two (2) multiplied by the number of points of the star in such case being that of six (6). Therefore AP is equal to 2p.

Find the expression for the 6-stellar number at stage S7:

Sn = S1 + [(n-1) ÷2] x [2 x 2P + (n – 1 -1) 2P]

Sn = 1 + [(n-1) ÷2] x [2 x 12 + (n – 2) 12]

S7 = 1 + [(7-1) ÷2] x [2 x 12 + (7– 2) 12]

S7 = 1 + [(6) ÷2] x [24 + (5) 12]

S7 = 1 + [3] x [24 + 60]

S7 = 1 + 3 x 84

S7 = 1 + 252

S7 = 253

Find a general statement for the 6-stellar number at stage Sn in terms of n:

Sn = S1 + [(n-1) ÷2] x [2 x 2P + (n – 1 -1) 2P]

Sn = S1 + [(n-1) ÷2] x [2 x 12 + (n – 2) 12]

Sn = S1 + [(n-1) ÷2] x [24 + (n – 2) 12]

Test the general statement for the 6-stellar numbers for multiple stages:

General Statement: Sn = S1 + [(n-1) ÷2] x [24 + (n – 2) 12]

Stage S3:

S3 = S1 + [(3-1) ÷2] x [24 + (3 – 2) 12]

S3 = 1 + [2 ÷2] x [24 + (1) 12]

S3 = 1 + [1] x [24 + 12]

S3 = 1 + [36]

S3 = 37

Using the general statement it is possible to calculate that S3 of the 6- Stellar star has 37 dots in it. From the Diagram of the 6 point Stellar Star it is possible to show the formula is true for S3.

Stage S4:

S4 = S1 + [(4-1) ÷2] x [24 + (4 – 2) 12]

S4 = 1 + [(3) ÷2] x [24 + (2) 12]

S4 = 1 + [(3) ÷2] x [24 + 24]

S4 = 1 + [(3) ÷2] x [48]

S4 = 1 + 72

S4 = 73

The general statement makes it is possible to calculate that S4 of the 6- Stellar star has 73 dots in it. The Diagram of the 6 point Stellar Star it is possible to show the formula is true for S4.

Stage S3:

S3 = S1 + [(n-1) ÷2] x [28 + (n – 2) 14]

S3 = 1 + [(3-1) ÷2] x [28 + (3 – 2) 14]

S3 = 1 + [2 ÷2] x [42]

S3 = 43

Using the general statement it is possible to calculate that S3 of the 7- Stellar star has 43 dots. By continuing diagram 5 it is possible to support this outcome of produced from the general statement.

Limitations:

There are many faults with this general statement. Although it applies to many different numbered points of Stellar Stars, for it to be applied it is necessary to determine that the star has the same formation as that of the star which the general statement was created for.  In order to determine this it is necessary to determine that the arithmetic progression is equal to double that of the number of points of the star.

The number of points or P cannot be a negative number as this is not possible to form a negative star shape. The stellar shapes also only work when the number of point or vertices are equal to or greater than three (3). Thus it can be concluded that the formula only works for all possible stellar formation shapes.

This general statement also assumes that the first stellar unit is equal to that of one. Meaning that if the sequence of numbers started at any other numerical value the general formula may not apply. Therefore if your sequence did not include the first few layers it is possible the formula would not work as the S1.