In mathematics, the Fibonacci numbers are the numbers in the following integer sequence: (sequence A000045 in OEIS). By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

Fn = Fn-1 + Fn-2

with seed values

F0 = 0, F1 = 1

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics. (By modern convention, the sequence begins with F0 = 0. The Liber Abaci began the sequence with F1 = 1, omitting the initial 0, and the sequence is still written this way by some.) Fibonacci numbers are closely related to Lucas numbers in that they are a complementary pair of Lucas sequences. They are intimately connected with the golden ratio, for example the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... . Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruit spouts of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone

The Fibonacci numbers:
F0F1F2F3F4F5F6F7F8F9F10F11F12F13F14F15F16F17F18F19F20F21F22F23F24F25Fn
0112358132134558914423337761098715972584418167651094617711286574636875025...
Interesting properties
- F sum identities:

F0 + F1 + F2 + F3 + ... + Fn = Fn+2 - 1

1F1 + 2F2 + 3F3 + ... + nFn = nFn+2 - Fn+3 + 2

- Lucas' theorem::

Fm gcd Fn = F(m gcd n)  where gcd = greatest common divisor
- Cassini's Formula:
Fn+1 · Fn-1 - (Fn)2 = (-1)n
- Variant of this formula: 
Fn-2 · Fn+1 - Fn-1 · Fn = (-1)n-1
- Simson's relation: 
Fn+1 · Fn-1 + (-1)n-1 = (Fn)2
- Shifting property: 
Fm+n = Fm · Fn+1 + Fm-1 · Fn