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| Meta Title | Fibonacci Sequence |
| Meta Description | The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding up the two numbers before it: |
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| Boilerpipe Text | The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
The next number is found by adding up the two numbers before it:
the 2 is found by adding the two numbers before it (1+1),
the 3 is found by adding the two numbers before it (1+2),
the 5 is (2+3),
and so on!
Example: the next number in the sequence above is 21+34
=
55
It is that simple!
Here is a longer list:
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, 121393, 196418, 317811, 514229, ...
Can you figure out the next few numbers?
Makes A Spiral
We van make squares with fibonacci side lengths, place them next to each other, and we get a nice spiral:
The Fibonacci Spiral
See how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.
This spiral is found in nature!
See:
Nature, The Golden Ratio, and Fibonacci
The Rule
The Fibonacci Sequence can be written as a "Rule" (see
Sequences and Series
).
First, the terms are numbered from 0 onwards like this:
n
=
x
n
=
0
0
1
1
2
1
3
2
4
3
5
5
6
8
7
13
8
21
9
34
10
55
11
89
12
144
13
233
14
377
...
So term number 6 is called
x
6
(which equals 8).
Example: the
8th
term is
the
7th
term plus the
6th
term:
x
8
=
x
7
+ x
6
So we can write the
rule
:
x
n
=
x
n−1
+ x
n−2
where:
x
n
is term number "n"
x
n−1
is the previous term (n−1)
x
n−2
is the term before that (n−2)
Example: term 9 is calculated like this:
x
9
= x
9−1
+ x
9−2
= x
8
+ x
7
= 21 + 13
= 34
Rules like this are called
recursive
formulas.
To use a recursive formula we also need to know the first few terms. For Fibonacci we start with x
0
=
0 and x
1
=
1
Golden Ratio
And here is a surprise.
When we take any two successive
(one after the other)
Fibonacci Numbers, their ratio is very close to the
Golden Ratio
"
φ
" which is approximately
1.618034...
The Golden Ratio is found in art, architecture, and nature.
In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:
a
b
b / a
2
3
1.5
3
5
1.666666666...
5
8
1.6
8
13
1.625
...
...
...
144
233
1.618055556...
233
377
1.618025751...
...
...
...
Try some start values yourself:
It can take longer to get good values, but it shows that not just the Fibonacci Sequence can do this!
Using The Golden Ratio to Calculate Fibonacci Numbers
And even more surprising is that we can
calculate any Fibonacci Number
using the Golden Ratio:
x
n
=
φ
n
− (1−φ)
n
√5
The answer comes out
as a whole number
, exactly equal to the addition of the previous two terms.
Example: x
6
x
6
=
(1.618034...)
6
− (1−1.618034...)
6
√5
When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer
8.00000033
, a more accurate calculation would be closer to 8.
Try n=12 and see what you get.
You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):
Example: What is the next in the sequence after 8 ?
It will be 8 times φ:
8φ
=
8 × 1.618034...
=
12.94427...
=
13
(rounded)
Some Interesting Things
An odd fact:
The sequence goes even,
odd
,
odd
, even,
odd
,
odd
, even,
odd
,
odd
, ... :
0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
...
Why?
Because adding
two odd numbers produces an even number,
but adding
even and odd (in any order) produces an odd number.
Lucas Numbers
Starting the sequence with
2
and
1
we get the "Lucas Numbers". They get closer and closer to the powers (
exponents
) of the Golden Ratio:
Lucas
Number
n
φ
n
2
0
1.0000...
1
1
1.6180...
3
2
2.6180...
4
3
4.2361...
7
4
6.8541...
11
5
11.0902...
18
6
17.9443...
29
7
29.0344...
47
8
46.9787...
76
9
76.0132...
123
10
122.9919...
199
11
199.0050...
...
...
...
For example, the
15th
Lucas Number is approximately φ
15
=
1364.0007..., so is exactly
1364
. Try it yourself!
Here is the Fibonacci sequence again:
n
=
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
...
x
n
=
0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
...
There is an interesting pattern:
Look at the number
x
3
=
2
. Every
3
rd number is a multiple of
2
(2, 8, 34,144,610, ...)
Look at the number
x
4
=
3
. Every
4
th number is a multiple of
3
(3, 21,144, ...)
Look at the number
x
5
=
5
. Every
5
th number is a multiple of
5
(5, 55,610, ...)
And so on (every
n
th number is a multiple of
x
n
).
1/89
=
0.011235955056179775...
Notice the first few digits (0, 1, 1, 2, 3, 5) are the Fibonacci sequence?
In a way they
all
are, except multiple digit numbers (13, 21, and so on)
overlap
, like this:
0.0
0.01
0.001
0.0002
0.00003
0.000005
0.0000008
0.00000013
0.000000021
... and so on ...
0.011235955056179775...
=
1/89
Fibonacci Words
I
have
a poem
about a sequence
that will not stop growin'
where each line has as many words as
the previous two lines combined which sounds real easy but can so rapidly
grow out of control until I have no idea how I can continue to meet the needs of
this Fibonacci thingy
by Rod Pierce
Terms Below Zero
The sequence works below zero also, like this:
n
=
...
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
...
x
n
=
...
−8
5
−3
2
−1
1
0
1
1
2
3
5
8
...
(Prove to yourself that each number is found by adding up the two numbers before it!)
In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a
+-+-
... pattern. It can be written like this:
x
−n
=
(−1)
n+1
x
n
Which says term "−n" is equal to
(−1)
n+1
times term "n", and the value
(−1)
n+1
neatly makes the correct +1, −1, +1, −1, ... pattern.
History
Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!
About Fibonacci The Man
His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.
"Fibonacci" was his nickname, which roughly means "Son of Bonacci".
As well as being famous for the Fibonacci Sequence, he helped spread
Hindu-Arabic Numerals
(like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of
Roman Numerals
(I, II, III, IV, V, and so on). That has saved us all a lot of trouble! Thank you Leonardo.
Fibonacci Day
Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. So next Nov 23 let everyone know! |
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# Fibonacci Sequence
The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
The next number is found by adding up the two numbers before it:
- the 2 is found by adding the two numbers before it (1+1),
- the 3 is found by adding the two numbers before it (1+2),
- the 5 is (2+3),
- and so on\!
Example: the next number in the sequence above is 21+34 *\=* **55**
It is that simple\!
Here is a longer list:
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, 121393, 196418, 317811, 514229, ...
*Can you figure out the next few numbers?*
## Makes A Spiral
We van make squares with fibonacci side lengths, place them next to each other, and we get a nice spiral:
**The Fibonacci Spiral**
See how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.
This spiral is found in nature\!
See: [Nature, The Golden Ratio, and Fibonacci](https://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html)
## The Rule
The Fibonacci Sequence can be written as a "Rule" (see [Sequences and Series](https://www.mathsisfun.com/algebra/sequences-series.html)).
First, the terms are numbered from 0 onwards like this:
n *\=* xn *\=*
0 0
1 1
2 1
3 2
4 3
5 5
6 8
7 13
8 21
9 34
10 55
11 89
12 144
13 233
14 377
...
So term number 6 is called x6 (which equals 8).
| | |
|---|---|
| Example: the **8th** term is the **7th** term plus the **6th** term: x8 *\=* x7 + x6 |  |
So we can write the **rule**:
**xn *\=* xn−1 + xn−2**
where:
- **xn** is term number "n"
- **xn−1** is the previous term (n−1)
- **xn−2** is the term before that (n−2)
### Example: term 9 is calculated like this:
x9\= x9−1 + x9−2\= x8 + x7\= 21 + 13 \= 34
Rules like this are called **recursive** formulas.
To use a recursive formula we also need to know the first few terms. For Fibonacci we start with x0 *\=* 0 and x1 *\=* 1
## Golden Ratio
And here is a surprise.
When we take any two successive *(one after the other)* Fibonacci Numbers, their ratio is very close to the [Golden Ratio](https://www.mathsisfun.com/numbers/golden-ratio.html) "**φ**" which is approximately **1\.618034...**
The Golden Ratio is found in art, architecture, and nature.
In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:
| a | b | | b / a |
|---|---|---|---|
| 2 | 3 | | 1\.5 |
| 3 | 5 | | 1\.666666666... |
| 5 | 8 | | 1\.6 |
| 8 | 13 | | 1\.625 |
| ... | ... | | ... |
| 144 | 233 | | 1\.618055556... |
| 233 | 377 | | 1\.618025751... |
| ... | ... | | ... |
Try some start values yourself:
images/fibo.js
It can take longer to get good values, but it shows that not just the Fibonacci Sequence can do this\!
## Using The Golden Ratio to Calculate Fibonacci Numbers
And even more surprising is that we can **calculate any Fibonacci Number** using the Golden Ratio:
xn *\=* φn − (1−φ)n *√5*
The answer comes out **as a whole number**, exactly equal to the addition of the previous two terms.
### Example: x6
x6 *\=* (1.618034...)6 − (1−1.618034...)6 *√5*
When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer **8\.00000033** , a more accurate calculation would be closer to 8.
Try n=12 and see what you get.
You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):
### Example: What is the next in the sequence after 8 ?
It will be 8 times φ:
8φ *\=* 8 × 1.618034...
*\=* 12.94427...
*\=* **13** (rounded)
## Some Interesting Things
### An odd fact:
The sequence goes even, **odd**, **odd**, even, **odd**, **odd**, even, **odd**, **odd**, ... :
| | | | | | | | | | | | | | | | | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | **1** | **1** | 2 | **3** | **5** | 8 | **13** | **21** | 34 | **55** | **89** | 144 | **233** | **377** | 610 | ... |
Why?
Because adding **two odd numbers produces an even number,** but adding **even and odd (in any order) produces an odd number.**
### Lucas Numbers
Starting the sequence with **2** and **1** we get the "Lucas Numbers". They get closer and closer to the powers ([exponents](https://www.mathsisfun.com/exponent.html)) of the Golden Ratio:
| Lucas Number | n | φn |
|---|---|---|
| **2** | 0 | 1\.0000... |
| **1** | 1 | 1\.6180... |
| 3 | 2 | 2\.6180... |
| 4 | 3 | 4\.2361... |
| 7 | 4 | 6\.8541... |
| 11 | 5 | 11\.0902... |
| 18 | 6 | 17\.9443... |
| 29 | 7 | 29\.0344... |
| 47 | 8 | 46\.9787... |
| 76 | 9 | 76\.0132... |
| 123 | 10 | 122\.9919... |
| 199 | 11 | 199\.0050... |
| ... | ... | ... |
For example, the **15th** Lucas Number is approximately φ15 *\=* 1364.0007..., so is exactly **1364**. Try it yourself\!
### Here is the Fibonacci sequence again:
| | | | | | | | | | | | | | | | | | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| *n *\=** | *0* | *1* | *2* | *3* | *4* | *5* | *6* | *7* | *8* | *9* | *10* | *11* | *12* | *13* | *14* | *15* | *...* |
| xn *\=* | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 | 377 | 610 | ... |
There is an interesting pattern:
- Look at the number **x3 *\=* 2**. Every **3**rd number is a multiple of **2** (2, 8, 34,144,610, ...)
- Look at the number **x4 *\=* 3**. Every **4**th number is a multiple of **3** (3, 21,144, ...)
- Look at the number **x5 *\=* 5**. Every **5**th number is a multiple of **5** (5, 55,610, ...)
And so on (every **n**th number is a multiple of **xn**).
### 1/89 *\=* 0.011235955056179775...
Notice the first few digits (0, 1, 1, 2, 3, 5) are the Fibonacci sequence?
In a way they **all** are, except multiple digit numbers (13, 21, and so on) **overlap**, like this:
| |
|---|
| 0\.0 |
| 0\.01 |
| 0\.001 |
| 0\.0002 |
| 0\.00003 |
| 0\.000005 |
| 0\.0000008 |
| 0\.00000013 |
| 0\.000000021 |
| ... and so on ... |
| **0\.011235955056179775...** *\=* 1/89 |
### Fibonacci Words
I
have
a poem
about a sequence
that will not stop growin'
where each line has as many words as
the previous two lines combined which sounds real easy but can so rapidly
grow out of control until I have no idea how I can continue to meet the needs of
this Fibonacci thingy
*by Rod Pierce*
## Terms Below Zero
The sequence works below zero also, like this:
| | | | | | | | | | | | | | | | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| *n *\=** | *...* | *−6* | *−5* | *−4* | *−3* | *−2* | *−1* | ***0*** | *1* | *2* | *3* | *4* | *5* | *6* | *...* |
| xn *\=* | ... | −8 | 5 | −3 | 2 | −1 | 1 | 0 | 1 | 1 | 2 | 3 | 5 | 8 | ... |
*(Prove to yourself that each number is found by adding up the two numbers before it!)*
In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a \+-+- ... pattern. It can be written like this:
**x**−n *\=* (−1)n+1 **x**n
Which says term "−n" is equal to (−1)n+1 times term "n", and the value (−1)n+1 neatly makes the correct +1, −1, +1, −1, ... pattern.
## History
Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before\!
## About Fibonacci The Man
His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.
"Fibonacci" was his nickname, which roughly means "Son of Bonacci".
As well as being famous for the Fibonacci Sequence, he helped spread [Hindu-Arabic Numerals](https://www.mathsisfun.com/place-value.html) (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of [Roman Numerals](https://www.mathsisfun.com/roman-numerals.html) (I, II, III, IV, V, and so on). That has saved us all a lot of trouble! Thank you Leonardo.
## Fibonacci Day
Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. So next Nov 23 let everyone know\!
Mathopolis:[Q1]() [Q2]() [Q3]() [Q4]() [Q5]() [Q6]() [Q7]() [Q8]() [Q9]() [Q10]()
[Golden Ratio](https://www.mathsisfun.com/numbers/golden-ratio.html) [Nature, Golden Ratio and Fibonacci Numbers](https://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html) [Number Patterns](https://www.mathsisfun.com/numberpatterns.html)
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Copyright © 2025 Rod Pierce |
| Readable Markdown | The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
The next number is found by adding up the two numbers before it:
- the 2 is found by adding the two numbers before it (1+1),
- the 3 is found by adding the two numbers before it (1+2),
- the 5 is (2+3),
- and so on\!
Example: the next number in the sequence above is 21+34 *\=* **55**
It is that simple\!
Here is a longer list:
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, 121393, 196418, 317811, 514229, ...
*Can you figure out the next few numbers?*
## Makes A Spiral
We van make squares with fibonacci side lengths, place them next to each other, and we get a nice spiral:
**The Fibonacci Spiral**
See how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.
This spiral is found in nature\!
See: [Nature, The Golden Ratio, and Fibonacci](https://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html)
## The Rule
The Fibonacci Sequence can be written as a "Rule" (see [Sequences and Series](https://www.mathsisfun.com/algebra/sequences-series.html)).
First, the terms are numbered from 0 onwards like this:
n *\=* xn *\=*
0 0
1 1
2 1
3 2
4 3
5 5
6 8
7 13
8 21
9 34
10 55
11 89
12 144
13 233
14 377
...
So term number 6 is called x6 (which equals 8).
| | |
|---|---|
| Example: the **8th** term is the **7th** term plus the **6th** term: x8 *\=* x7 + x6 |  |
So we can write the **rule**:
**xn *\=* xn−1 + xn−2**
where:
- **xn** is term number "n"
- **xn−1** is the previous term (n−1)
- **xn−2** is the term before that (n−2)
### Example: term 9 is calculated like this:
x9\= x9−1 + x9−2\= x8 + x7\= 21 + 13 \= 34
Rules like this are called **recursive** formulas.
To use a recursive formula we also need to know the first few terms. For Fibonacci we start with x0 *\=* 0 and x1 *\=* 1
## Golden Ratio
And here is a surprise.
When we take any two successive *(one after the other)* Fibonacci Numbers, their ratio is very close to the [Golden Ratio](https://www.mathsisfun.com/numbers/golden-ratio.html) "**φ**" which is approximately **1\.618034...**
The Golden Ratio is found in art, architecture, and nature.
In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:
| a | b | | b / a |
|---|---|---|---|
| 2 | 3 | | 1\.5 |
| 3 | 5 | | 1\.666666666... |
| 5 | 8 | | 1\.6 |
| 8 | 13 | | 1\.625 |
| ... | ... | | ... |
| 144 | 233 | | 1\.618055556... |
| 233 | 377 | | 1\.618025751... |
| ... | ... | | ... |
Try some start values yourself:
It can take longer to get good values, but it shows that not just the Fibonacci Sequence can do this\!
## Using The Golden Ratio to Calculate Fibonacci Numbers
And even more surprising is that we can **calculate any Fibonacci Number** using the Golden Ratio:
xn *\=* φn − (1−φ)n *√5*
The answer comes out **as a whole number**, exactly equal to the addition of the previous two terms.
### Example: x6
x6 *\=* (1.618034...)6 − (1−1.618034...)6 *√5*
When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer **8\.00000033** , a more accurate calculation would be closer to 8.
Try n=12 and see what you get.
You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):
### Example: What is the next in the sequence after 8 ?
It will be 8 times φ:
8φ *\=* 8 × 1.618034...
*\=* 12.94427...
*\=* **13** (rounded)
## Some Interesting Things
### An odd fact:
The sequence goes even, **odd**, **odd**, even, **odd**, **odd**, even, **odd**, **odd**, ... :
| | | | | | | | | | | | | | | | | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | **1** | **1** | 2 | **3** | **5** | 8 | **13** | **21** | 34 | **55** | **89** | 144 | **233** | **377** | 610 | ... |
Why?
Because adding **two odd numbers produces an even number,** but adding **even and odd (in any order) produces an odd number.**
### Lucas Numbers
Starting the sequence with **2** and **1** we get the "Lucas Numbers". They get closer and closer to the powers ([exponents](https://www.mathsisfun.com/exponent.html)) of the Golden Ratio:
| Lucas Number | n | φn |
|---|---|---|
| **2** | 0 | 1\.0000... |
| **1** | 1 | 1\.6180... |
| 3 | 2 | 2\.6180... |
| 4 | 3 | 4\.2361... |
| 7 | 4 | 6\.8541... |
| 11 | 5 | 11\.0902... |
| 18 | 6 | 17\.9443... |
| 29 | 7 | 29\.0344... |
| 47 | 8 | 46\.9787... |
| 76 | 9 | 76\.0132... |
| 123 | 10 | 122\.9919... |
| 199 | 11 | 199\.0050... |
| ... | ... | ... |
For example, the **15th** Lucas Number is approximately φ15 *\=* 1364.0007..., so is exactly **1364**. Try it yourself\!
### Here is the Fibonacci sequence again:
| | | | | | | | | | | | | | | | | | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| *n *\=** | *0* | *1* | *2* | *3* | *4* | *5* | *6* | *7* | *8* | *9* | *10* | *11* | *12* | *13* | *14* | *15* | *...* |
| xn *\=* | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 | 377 | 610 | ... |
There is an interesting pattern:
- Look at the number **x3 *\=* 2**. Every **3**rd number is a multiple of **2** (2, 8, 34,144,610, ...)
- Look at the number **x4 *\=* 3**. Every **4**th number is a multiple of **3** (3, 21,144, ...)
- Look at the number **x5 *\=* 5**. Every **5**th number is a multiple of **5** (5, 55,610, ...)
And so on (every **n**th number is a multiple of **xn**).
### 1/89 *\=* 0.011235955056179775...
Notice the first few digits (0, 1, 1, 2, 3, 5) are the Fibonacci sequence?
In a way they **all** are, except multiple digit numbers (13, 21, and so on) **overlap**, like this:
| |
|---|
| 0\.0 |
| 0\.01 |
| 0\.001 |
| 0\.0002 |
| 0\.00003 |
| 0\.000005 |
| 0\.0000008 |
| 0\.00000013 |
| 0\.000000021 |
| ... and so on ... |
| **0\.011235955056179775...** *\=* 1/89 |
### Fibonacci Words
I
have
a poem
about a sequence
that will not stop growin'
where each line has as many words as
the previous two lines combined which sounds real easy but can so rapidly
grow out of control until I have no idea how I can continue to meet the needs of
this Fibonacci thingy
*by Rod Pierce*
## Terms Below Zero
The sequence works below zero also, like this:
| | | | | | | | | | | | | | | | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| *n *\=** | *...* | *−6* | *−5* | *−4* | *−3* | *−2* | *−1* | ***0*** | *1* | *2* | *3* | *4* | *5* | *6* | *...* |
| xn *\=* | ... | −8 | 5 | −3 | 2 | −1 | 1 | 0 | 1 | 1 | 2 | 3 | 5 | 8 | ... |
*(Prove to yourself that each number is found by adding up the two numbers before it!)*
In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a \+-+- ... pattern. It can be written like this:
**x**−n *\=* (−1)n+1 **x**n
Which says term "−n" is equal to (−1)n+1 times term "n", and the value (−1)n+1 neatly makes the correct +1, −1, +1, −1, ... pattern.
## History
Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before\!
## About Fibonacci The Man
His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.
"Fibonacci" was his nickname, which roughly means "Son of Bonacci".
As well as being famous for the Fibonacci Sequence, he helped spread [Hindu-Arabic Numerals](https://www.mathsisfun.com/place-value.html) (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of [Roman Numerals](https://www.mathsisfun.com/roman-numerals.html) (I, II, III, IV, V, and so on). That has saved us all a lot of trouble! Thank you Leonardo.
## Fibonacci Day
Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. So next Nov 23 let everyone know\! |
| Shard | 29 (laksa) |
| Root Hash | 10041072732158755029 |
| Unparsed URL | com,mathsisfun!www,/numbers/fibonacci-sequence.html s443 |