# integers.info

## Fibonacci

### F0 = 0

The initial number in the Fibonacci sequence is 0.

Drag the slider to the right to visualize the sequnce.

The Fibonacci numbers are a sequence of integers defined as:

F0 = 0
F1 = 1
Fn = Fn − 1 + Fn − 2

The first two numbers are 0 and 1, and thereafter, every number is equal to the sum of the two previous numbers. This is illustrated above where the side of each square is equal to the sides of two previous squares combined. When a spiral is drawn using circular arcs across each square, it is called the Fibonacci Spiral.

The Fibonacci sequence is named after Medieval mathematician Leonardo Fibonacci, who popularized the number sequence in his book Liber Abaci in the early 13th century. He used the Fibonacci sequence to predict the population growth of breeding rabbits. The Fibonacci numbers appear frequently in nature, for example in the petal leaves of flowers and in the spiral shape of shells.

### The Golden Ratio

The Golden Ratio, φ, is defined as:

 a + b = a = φ ≈ 1.6180339887 a b

This means that when the ratio between a and b is the same as the ratio between a + b and a, it is the Golden Ratio. The Fibonacci sequence is heavily interconnected with the Golden Ratio, since the ratio between neighboring Fibonacci numbers gets closer and closer to the Golden Ratio, the higher the numbers get.

### List of Fibonacci Numbers

Here are all the Fibonacci numbers smaller than one quadrillion:

IndexFibonacci Number
F00
F11
F21
F32
F43
F55
F68
F713
F821
F934
F1055
F1189
F12144
F13233
F14377
F15610
F16987
F171597
F182584
F194181
F206765
F2110946
F2217711
F2328657
F2446368
F2575025
F26121393
F27196418
F28317811
F29514229
F30832040
F311346269
F322178309
F333524578
F345702887
F359227465
F3614930352
F3724157817
F3839088169
F3963245986
F40102334155
F41165580141
F42267914296
F43433494437
F44701408733
F451134903170
F461836311903
F472971215073
F484807526976
F497778742049
F5012586269025
F5120365011074
F5232951280099
F5353316291173
F5486267571272
F55139583862445
F56225851433717
F57365435296162
F58591286729879
F59956722026041
F601548008755920
F612504730781961
F624052739537881
F636557470319842
F6410610209857723
F6517167680177565
F6627777890035288
F6744945570212853
F6872723460248141
F69117669030460994
F70190392490709135
F71308061521170129
F72498454011879264
F73806515533049393