Liber Quadratorum

Liber Quadratorum is recognized as Fibonacci's most impressive collection of knowledge. It was written in 1225 and dealt with number theory. One interesting thing contained in his book is the equation he came up with for finding square numbers (1, 4, 9, 16, 25, etc). He realized that each square number was the sum of the previous square and some odd number:

For some n in the whole numbers: n²+(2n+1)=(n+1)² 

Let n=1,  1+(3)=4. 

Let n=2, 4+(5)=9. 

Let n=3, 9+(7)=16.

Another formula Fibonacci found dealt with finding Pythagorean Triples. He would first find an odd, square number. Then he would add up all the odd numbers from 1 to the chosen number, including one, but excluding the latter. Taking the square of that sum would produce the second square. Adding the two square numbers together would give him the square number whose square would produce the final part of the Pythagorean Triple. I included an example below.
Choose 5 as one of the first squares.

5²=25...5 is the first part of the Triple.

The odd numbers from 1 up to but not including 25 are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23.

The sum of these numbers is 144.
sqrt(144)=12<--- second part

144+25=169

sqrt(169)=13<--- third part

Therefore the Pythagorean triple is 5, 12, 13.

http://www.bing.com/images/search?q=pythagorean+triple&FORM=HDRSC2#view=detail&id=E8B31E2011B18D1FD0CDABE2BF67A9A48BCE68A4&selectedIndex=9

Sources:
O'Connor, J. J. "Leonardo Pisano Fibonacci." Fibonacci Biography. N.p., n.d. Web. 02 Mar. 2014.