One of the numbers in the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,…, where each number (after the second) is the sum of the preceding two. This sequence has many interesting properties. For instance, the sequence consisting of the ratios of one Fibonacci number to the previous one, $\frac{1}{1},\frac{2}{1},\frac{3}{2},\frac{5}{3},\frac{8}{5},\frac{13}{8},\dots$, has the limit φ‎ the golden ratio. Binet’s theorem states that the nth Fibonacci number equals $\left({\mathrm{\phi }}^n-{\left(-\mathrm{\phi }\right)}^{-n}\right)/\sqrt{5}$. See also difference equation, generating function, Lucas numbers.

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html A site with examples of Fibonacci numbers in nature and the relationship to the golden section.