# fibonacci

Liber Abaci, Leonardo di Pisa (1202)

The Liber Abaci, written in 1202 by Leonardo di Pisa, more popularly known as Fibonacci (1180-1250), was the first European publication using Hindu-Arabic numerals.  It also contained a problem related to the regeneration of rabbits that produces a sequence of numbers, called the Fibonacci Sequence:

“How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair, which becomes productive from the second  month on?”

Start of Month 1:                 0 Adult pair                        1 Baby pair                     Total =  1

Start of Month 2:                1 Adult pair                         0 Baby pair                                  1

Start of Month 3:                1 Adult pair                        1 Baby pair                                   2

Start of Month 4:                2 Adult pair                        1 Baby pair                                   3

Start of Month 5:                3 Adult pair                        2 Baby pair                                   5

Start of Month 6:                5 Adult pair                        3 Baby pair                                   8

etc…

This is an example of a recurrence relationship, and is found in dynamical systems such as the Mandelbrot Set which describes fractal patterns in Nautre.  It is an equation which defines the system recursively, that is, each term of the sequence is defined as a function of the preceding terms:

Fn = Fn – 1  + Fn – 2

The first 16 Fibonacci Numbers:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610

Every counting number is either a Fibonacci Number or can be expressed as a sum of non-consecutive Fibonacci Numbers…

51 = 34 + 17; 17 = 13 + 4 ; 4 = 3 + 1 ; so, 51 = 34 + 13 + 3 + 1

Furthermore, (Fn)2 – (Fn+1)(Fn-1) = ± 1     examples:  132 – (8)(21) = 1;  212 – (13)(34) =  -1

phi identities showing fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.:

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