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Number Series

There are all kinds of numerical series.

Series at a Glance

Counting	1+1+1+1+1+...	1, 2, 3, 4, 5,...
Triangular	1+2+3+4+5+...	1, 3, 6, 10, 15,...
Square	1+3+5+7+9+...	1, 4, 9, 16, 25,...
Pentagonal	1+4+7+10+13+...	1, 5, 12, 22, 35,...
Hexagonal	1+5+9+13+17+...	1, 6, 15, 28, 45,...
Heptagonal	1+6+11+16+21+...	1, 7, 18, 34, 55,...
Octagonal	1+7+13+19+25+...	1, 8, 21, 40, 65,...
Fibonacci	1+1,1+2,2+3,3+5,5+8,...	2, 3, 5, 8, 13,...
Tetrahedral	1+3+6+10+15+...	1, 4, 10, 20, 35,...	(1/6)n(n+1)(n+2)
Square Pyramid	12+22+32+42+52+...	1, 5, 14, 30, 55,...	(1/6)n(n+1)(2n+1)
Squares	12, 22, 32, 42, 52,...	1, 4, 9, 16, 25,...
Cubes	13, 23, 33, 43, 53,...	1, 8, 27, 64, 125,...
4-D Pantatope	1+4+10+20+35+...	1, 5, 15, 35, 70,,...

Natural Series

The most basic one is the Natural series, where all the numbers are in counting order like this:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,...

Every number is higher than the previous by the sum of 1, that is, 1 + 1 = 2, 4 + 1 = 5, etc., like this:

1 ⁺¹⁼ 2 ⁺¹⁼ 3 ⁺¹⁼ 4 ⁺¹⁼ 5 ⁺¹⁼ 6 ⁺¹⁼ 7 ⁺¹⁼ 8 ⁺¹⁼ 9 ⁺¹⁼ 10 ⁺¹⁼ 11 ⁺¹⁼ 12 ⁺¹⁼...

Fibonacci Series

There is also the Fibonaci series. This is where you get another sum by adding the two previous numbers like this:

1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, etc.

To put it this way:

1 ⁺¹⁼ 2 ⁺¹⁼ 3 ⁺²⁼ 5 ⁺³⁼ 8 ⁺⁵⁼ 13 ⁺⁸⁼ 21 ⁺¹³⁼ 34 ⁺²¹⁼ 55 ⁺³⁴⁼ 89 ⁺⁵⁵⁼ 144...

Furthermore, the ratios between the two Fibonacci numbers get closer to a certain ratio: 3 to 5 = 0.6
5 to 8 = 0.625
8 to 13 = 0.6153846
13 to 21 = 0.6190476
21 to 34 = 0.617647
34 to 55 = 0.6181818
55 to 89 = 0.6179775
89 to 144 = 0.6180555
144 to 233 = 0.6180257
233 to 377 = 0.6180371
377 to 610 = 0.6180327
610 to 987 = 0.6180344
987 to 1597 = 0.6180338
1597 to 2584 = 0.6180340
...

The number is equlivalent of (SQR(5)-1)/2

The Fibonacci Golden Number is approximately 0.6180340

Amazing about the fact is that 1 + 0.6180340 = 1/0.6180340 = 1.6180340

Also, 0.6180340 + 0.6180340² = 1

Triangular Series

This is a series where you take two natural numbers in a series and create another number and so forth to create a new series like this:

Natural Series: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...
1 ⁺²⁼ 3 ⁺³⁼ 6 ⁺⁴⁼ 10 ⁺⁵⁼ 15 ⁺⁶⁼ 21 ⁺⁷⁼ 28 ⁺⁸⁼ 36 ⁺⁹⁼ 45 ⁺¹⁰⁼ 55 ⁺¹¹⁼ 66 ⁺¹²⁼ 78 ⁺¹³⁼ 91 ⁺¹⁴⁼ 105 ⁺¹⁵⁼ 120...

Pyramid Series

This series takes the previous Triangular series and uses it as the basis for another series.

Triangular Series: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120,...

1 ⁺³⁼ 4 ⁺⁶⁼ 10 ⁺¹⁰⁼ 20 ⁺¹⁵⁼ 35 ⁺²¹⁼ 56 ⁺²⁸⁼ 84 ⁺³⁶⁼ 120 ⁺⁴⁵⁼ 165 ⁺⁵⁵⁼ 220...

The Square Pyramid Series takes the list you created a bit further...

1 ⁺⁴⁼ 5 ⁺⁹⁼ 14 ⁺¹⁶⁼ 30 ⁺²⁵⁼ 55 ⁺³⁶⁼ 91 ⁺⁴⁵⁼ 136...

Triangular Table of Series

Check out this table. There are additional series such as 4-D Tetrahedral and others you can figure out on your own.

just ones --																													1
Natural --																												1		1
Triangular --																											1		2		1
Tetrahedal --																										1		3		3		1
4-D Tetrahedal --																									1		4		6		4		1
5-D Tetrahedal --																								1		5		10		10		5		1
																							1		6		15		20		15		6		1
																						1		7		21		35		35		21		7		1
																					1		8		28		56		70		56		28		8		1
																				1		9		36		84		126		126		84		36		9		1

Quadrigonal Numbers (Square)

This series is created by adding two immediate odd numbers together to get another number:

Odd: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21,...

1 ⁺³⁼ 4 ⁺⁵⁼ 9 ⁺⁷⁼ 16 ⁺⁹⁼ 25 ⁺¹¹⁼ 36 ⁺¹³⁼ 49 ⁺¹⁵⁼ 64 ⁺¹⁷⁼ 81 ⁺¹⁹⁼ 100 ⁺²¹⁼ 121...

You may also notice that the Quad series are squares of the Natural numbers series:

1², 2², 3², 4², 5², 6², 7², 8², 9², 10², 11²...

The sum of the first "n" odd numbers is a perfect square.

1+3+5+7+...+(2n-1)=n²

Pentagonal Numbers

Series Used to Create: 1, 4, 7, 10, 13, 16,...

1, 5, 12, 22, 35, 51,...

Hexagonal Numbers

Series used to create: 1, 5, 12, 22, 35, 51...

1, 6, 18, 40, 75, 126,...

Oblong Series

Series of 1x2, 2x3, 3x4, 4x5, 5x6, 6x7,...

2, 6, 12, 20, 30, 42,...

Note that they're the doubling of the triangular series 1, 3, 6, 10,...

Other Odd Series

The sum of the first "n" cubes is a square.

1³+ 2³= 9 = 3²

1³+ 2³+ 3³= 36 = 6²

1³+ 2³+ 3³+ 4³= 100 = 10²

1³+ 2³+ 3³+ 4³+ 5³= 225 = 15²

1³+ 2³+ 3³+ ... n³ = (1+2+3+...n)²

1+2+1=2²

1+2+3+2+1=3²

Harmonic Triangle

1	2	3	4	5	6
2	6	12	20	30
3	12	30	60
4	20	60
5	30
6

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