# Patterns & Sequences

Fibonacci Sequence
The numbers of the Fibonacci Sequence can either start 0,1, or 1,1, depending on the author. However, beyond that everyone agrees, so: 0, 1, 1, 2, 3, 5, 8, 13, ... With this sequence we add the 2 previous numbers to get the current number.
Do not let the name fool you, it was not 'invented' by Fibonacci. The sequence has been used for centuries before him. Many contributors to the topic came after Fibonacci as well.
While the sites listed have some good information, a few of them put too much emphasis on 'the rabbit problem', which was just a word problem to explain the sequence.
http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm
http://www.inspirationgreen.com/fibonacci-sequence-in-nature
https://math.temple.edu/~reich/Fib/fibo.html
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#golden
http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm
http://www.inspirationgreen.com/fibonacci-sequence-in-nature
http://science.howstuffworks.com/math-concepts/fibonacci-nature2.htm
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibBio.html
http://www.livescience.com/37470-fibonacci-sequence.html
http://www.goldennumber.net/art-composition-design/

Golden Ratio is related to the Fibonacci Sequence. In fact it can be found in the ratio of the sequences terms. However, it occurs in many other places as well.
Other Related topics include the Golden Rectangle and the Golden Spiral.
All of the related topics can found throughout Art & Nature, as well as many other places with very practical applications.
http://www.livescience.com/37704-phi-golden-ratio.html
www.faena.com/aleph/articles/what-does-phi-the-golden-ratio-sounds-like-listen/
http://www.goldennumber.net/nature/

General & Other
Making a pattern can be quite easy, anything that repeats could be a pattern, such as 565656, or cat dog cat dog cat dog.
Sequences are ordered sets of items, which makes it possible to easily determine the next item. This term is often reserved for numbers, but it very similar to a pattern.
For example, we can have a sequence starting with 2, 4, 6, and we add all 3 numbers to get the next, so the next number would be 12, i.e. 2, 4, 6, 12, 22, 40, ...
Series is often confused with a sequences, but mathematically it is the sum of the sequence.

Some good books
These will be added as time permits

Additional information on these topics can be found on the following sites, as well as many others:
Several Sites are listed in the appropriate sections.
More will be added as time permits

Definitions and Formula are from numerous years of teaching the topics, but have recently been double checked with:
Borowski, E. J., & Borwein, J. M. (2006). Collins Web-linked dictionary of mathematics. New York, NY: HarperCollins Pub.