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:
F0 | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | F11 | F12 | F13 | F14 | F15 | F16 | F17 | F18 | F19 | F20 | F21 | F22 | F23 | F24 | F25 | … | Fn |
0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 | 377 | 610 | 987 | 1597 | 2584 | 4181 | 6765 | 10946 | 17711 | 28657 | 46368 | 75025 | ... |  |
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