Fibonacci Series
The Fibonacci Series is a set of numbers that regularly occur in nature. They can be calculated by starting with the numbers 0 and 1. Keep adding the last two numbers to find the next number in the series.
example
0,1 (0 + 1 = 1) so next number is 1
0,1,1 (1 + 1 = 2) so next number is 2
0,1,1,2 (1 + 2 = 3) so next number is 3
0,1,1,2,3 (2 + 3 = 5) so next number is 5
0,1,1,2,3,5 .......
 
Here are the first few numbers in the Fibonacci Series
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, ....

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Lucas Series
There is another similar series of numbers called Lucas Numbers. These are calculated in exactly the same way, but you start with 2,1 (instead of 0,1)
Here are the first few numbers in the Lucas Series
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, ....
 

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Pascal's Triangle
Each number in the Pascal Triangle is calculated by adding together the two numbers directly above it. (Every line starts and ends with 1.)
The numbers in the Pascal Triangle appear in calculations like
(x + 1)ⁿ
e.g. (x + 1)⁶ uses the numbers from the last line of the triangle
(x + 1)⁶ = 1x⁶ + 6x⁵ + 15x⁴ + 20x³ + 15x² + 6x + 1

						1
					1		1
				1		2		1
			1		3		3		1
		1		4		6		4		1
	1		5		10		10		5		1
1		6		15		20		15		6		1

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Triangular Number Series
Start with the number 0. Add 1 for the next number, add 2 for the next number, add 3 ...
You now get 0, 0+1, 0+1+2, 0+1+2+3, ....
The Triangular series starts
0, 1, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, ...