1 is also a special number because it is a square number as well. In fact, if you look at all of the regular polygonal figurative families, one is the first number. It is the first pentagonal number, hexagonal number, and so on. In fact, a dot can represent whatever figure you want it to, which shows the creativity of mathematics.
Because of this specialness to the number one, I thought we should analyze all of the regular polygonal figurative families. Let's look at the first fifteen numbers in each family.
Name

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Triangular

1

3

6

10

15

21

28

36

45

55

66

78

91

105

120

Square

1

4

9

16

25

36

49

64

81

100

121

144

169

196

225

Pentagonal

1

5

12

22

35

51

70

92

117

145

176

210

247

287

330

Hexagonal

1

6

15

28

45

66

91

120

153

190

231

276

325

378

435

Heptagonal

1

7

18

34

55

81

112

148

189

235

286

342

403

469

540

Octagonal

1

8

21

40

65

96

133

176

225

280

341

408

481

560

645

Nonagonal

1

9

24

46

75

111

154

204

261

325

396

474

559

651

750

Decagonal

1

10

27

52

85

126

175

232

297

370

451

540

637

742

855

This table just looks like a regular table of numbers. However, there are tons of patterns buried inside of it.
For example, look at each row. If you take the common differences of each one, what do you find?
The very top row has a constant difference of one, as you would expect. The next row, which contains the triangular numbers, has a difference of all natural numbers. The 1 and 3 have a difference of 2, the 3 and 6 have a difference of 3, the 6 and 10 have a difference of 4, and so on. In other words, rows n1 and n have a difference of n. This is the definition of triangular numbers, so that point is obvious.
The next row, which contains the square numbers, has a difference of all odd numbers. The 1 and 4 have a difference of 3, the 4 and 9 have a difference of 5, the 9 and 16 have a difference of 7, and so on. In other words, rows n1 and n have a difference of 2n1. If you want to see a quick proof of it, it isn't too hard. It just requires some algebra. Just do n squared minus n1 squared, and you should receive 2n1 to prove it.
n^{2}  (n  1)^{2} = 2n  1
n^{2 }  (n^{2}  2n + 1) = 2n  1n^{2 } n^{2} + 2n  1 = 2n  1
2n  1 = 2n  1
If you look at the pentagonal numbers, it isn't as obvious what the pattern is. However, the differences are 4, 7, 10, 13, 16, and so on. The differences all differ by three, making the equation 3n2. This can be proven using similar logic to above.
So, the first row has a difference of 1, which can be rewritten as 0n  (1). The second row has difference of n, or 1n  0. The third row has 2n  1, the fourth row has 3n  2. Do you see the pattern?
0n  (1)
1n  0
2n  1
3n  2
The fifth row of hexagonal numbers should have a difference of 4n  3 if this pattern continues, right? Let's see if it worked.
1  0 = 1 = 4(1)  3
6  1 = 5 = 4(2)  3
15  6 = 9 = 4(3)  3
28  15 = 13 = 4(4)  3
45  28 = 17 = 4(5)  3
The pattern indeed continued. For heptagonal numbers, the difference is 5n  4, for octagonal numbers, the difference is 6n  5, and so on.
Instead of looking at the rows this time, let's look at the columns and their differences. The first column has a difference of zero obviously. That is the nature of why one is in every figurative family. The second column has a difference of one. Since this number is the number defining the family itself, that isn't too surprising.
The third column has a difference of three. This is a little strange, but we can keep going. The fourth column has a difference of six. The fifth column has a difference of ten. Now do you see the pattern?
0, 1, 3, 6, 10
These are all triangular numbers! In fact, the triangular numbers have come back in the whole figurative family.
I find this table really cool to analyze. On June 15th (15 is a triangular number), we will revisit this table and look at some other neat identities within it.
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