<s>
In	O
mathematics	O
,	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
is	O
a	O
sequence	O
in	O
which	O
each	O
number	O
is	O
the	O
sum	O
of	O
the	O
two	O
preceding	O
ones	O
.	O
</s>
<s>
Numbers	O
that	O
are	O
part	O
of	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
are	O
known	O
as	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
commonly	O
denoted	O
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
were	O
first	O
described	O
in	O
Indian	O
mathematics	O
,	O
as	O
early	O
as	O
200	O
BC	O
in	O
work	O
by	O
Pingala	O
on	O
enumerating	O
possible	O
patterns	O
of	O
Sanskrit	O
poetry	O
formed	O
from	O
syllables	O
of	O
two	O
lengths	O
.	O
</s>
<s>
Fibonacci	B-Algorithm
numbers	I-Algorithm
appear	O
unexpectedly	O
often	O
in	O
mathematics	O
,	O
so	O
much	O
so	O
that	O
there	O
is	O
an	O
entire	O
journal	O
dedicated	O
to	O
their	O
study	O
,	O
the	O
Fibonacci	O
Quarterly	O
.	O
</s>
<s>
Applications	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
include	O
computer	O
algorithms	O
such	O
as	O
the	O
Fibonacci	B-Algorithm
search	I-Algorithm
technique	I-Algorithm
and	O
the	O
Fibonacci	B-Application
heap	I-Application
data	B-General_Concept
structure	I-General_Concept
,	O
and	O
graphs	O
called	O
Fibonacci	B-Algorithm
cubes	I-Algorithm
used	O
for	O
interconnecting	O
parallel	O
and	O
distributed	O
systems	O
.	O
</s>
<s>
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
also	O
strongly	O
related	O
to	O
the	O
golden	O
ratio	O
:	O
Binet	O
's	O
formula	O
expresses	O
the	O
th	O
Fibonacci	B-Algorithm
number	I-Algorithm
in	O
terms	O
of	O
and	O
the	O
golden	O
ratio	O
,	O
and	O
implies	O
that	O
the	O
ratio	O
of	O
two	O
consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
tends	O
to	O
the	O
golden	O
ratio	O
as	O
increases	O
.	O
</s>
<s>
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
also	O
closely	O
related	O
to	O
Lucas	B-Algorithm
numbers	I-Algorithm
,	O
which	O
obey	O
the	O
same	O
recurrence	O
relation	O
and	O
with	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
form	O
a	O
complementary	O
pair	O
of	O
Lucas	B-Algorithm
sequences	I-Algorithm
.	O
</s>
<s>
The	O
first	O
20	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
:	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
appears	O
in	O
Indian	O
mathematics	O
,	O
in	O
connection	O
with	O
Sanskrit	O
prosody	O
.	O
</s>
<s>
Counting	O
the	O
different	O
patterns	O
of	O
successive	O
L	O
and	O
S	O
with	O
a	O
given	O
total	O
duration	O
results	O
in	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
:	O
the	O
number	O
of	O
patterns	O
of	O
duration	O
units	O
is	O
.	O
</s>
<s>
Knowledge	O
of	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
was	O
expressed	O
as	O
early	O
as	O
Pingala	O
(	O
450BC	O
–	O
200BC	O
)	O
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
first	O
appears	O
in	O
the	O
book	O
Liber	O
Abaci	O
(	O
The	O
Book	O
of	O
Calculation	O
,	O
1202	O
)	O
by	O
Fibonacci	O
where	O
it	O
is	O
used	O
to	O
calculate	O
the	O
growth	O
of	O
rabbit	B-Algorithm
populations	O
.	O
</s>
<s>
Fibonacci	O
considers	O
the	O
growth	O
of	O
an	O
idealized	O
(	O
biologically	O
unrealistic	O
)	O
rabbit	B-Algorithm
population	O
,	O
assuming	O
that	O
:	O
a	O
newly	O
born	O
breeding	O
pair	O
of	O
rabbits	B-Algorithm
are	O
put	O
in	O
a	O
field	O
;	O
each	O
breeding	O
pair	O
mates	O
at	O
the	O
age	O
of	O
one	O
month	O
,	O
and	O
at	O
the	O
end	O
of	O
their	O
second	O
month	O
they	O
always	O
produce	O
another	O
pair	O
of	O
rabbits	B-Algorithm
;	O
and	O
rabbits	B-Algorithm
never	O
die	O
,	O
but	O
continue	O
breeding	O
forever	O
.	O
</s>
<s>
At	O
the	O
end	O
of	O
the	O
th	O
month	O
,	O
the	O
number	O
of	O
pairs	O
of	O
rabbits	B-Algorithm
is	O
equal	O
to	O
the	O
number	O
of	O
mature	O
pairs	O
(	O
that	O
is	O
,	O
the	O
number	O
of	O
pairs	O
in	O
month	O
)	O
plus	O
the	O
number	O
of	O
pairs	O
alive	O
last	O
month	O
(	O
month	O
)	O
.	O
</s>
<s>
The	O
number	O
in	O
the	O
th	O
month	O
is	O
the	O
th	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
The	O
name	O
"	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
"	O
was	O
first	O
used	O
by	O
the	O
19th-century	O
number	O
theorist	O
Édouard	O
Lucas	O
.	O
</s>
<s>
Like	O
every	O
sequence	O
defined	O
by	O
a	O
linear	O
recurrence	O
with	O
constant	O
coefficients	O
,	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
have	O
a	O
closed-form	O
expression	O
.	O
</s>
<s>
If	O
and	O
are	O
chosen	O
so	O
that	O
and	O
then	O
the	O
resulting	O
sequence	O
must	O
be	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
it	O
can	O
be	O
found	O
by	O
rounding	B-Algorithm
,	O
using	O
the	O
nearest	B-Algorithm
integer	I-Algorithm
function	I-Algorithm
:	O
</s>
<s>
In	O
fact	O
,	O
the	O
rounding	B-Algorithm
error	O
is	O
very	O
small	O
,	O
being	O
less	O
than	O
0.1	O
for	O
,	O
and	O
less	O
than	O
0.01	O
for	O
.	O
</s>
<s>
Fibonacci	B-Algorithm
numbers	I-Algorithm
can	O
also	O
be	O
computed	O
by	O
truncation	B-Algorithm
,	O
in	O
terms	O
of	O
the	O
floor	O
function	O
:	O
</s>
<s>
As	O
the	O
floor	O
function	O
is	O
monotonic	O
,	O
the	O
latter	O
formula	O
can	O
be	O
inverted	O
for	O
finding	O
the	O
index	O
of	O
the	O
smallest	O
Fibonacci	B-Algorithm
number	I-Algorithm
that	O
is	O
not	O
less	O
than	O
a	O
positive	O
integer	O
:	O
</s>
<s>
As	O
a	O
consequence	O
,	O
for	O
every	O
integer	O
d	O
>	O
1	O
there	O
are	O
either	O
4	O
or	O
5	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
with	O
d	O
decimal	O
digits	O
.	O
</s>
<s>
Johannes	O
Kepler	B-General_Concept
observed	O
that	O
the	O
ratio	O
of	O
consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
converges	B-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
,	O
because	O
the	O
ratios	O
between	O
consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
approaches	O
.	O
</s>
<s>
The	O
resulting	O
recurrence	O
relationships	O
yield	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
as	O
the	O
linear	O
coefficients	O
:	O
</s>
<s>
This	O
equation	O
can	O
be	O
proved	O
by	B-Algorithm
induction	I-Algorithm
on	O
:	O
</s>
<s>
Binet	O
's	O
formula	O
provides	O
a	O
proof	O
that	O
a	O
positive	O
integer	O
x	O
is	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
if	O
and	O
only	O
if	O
at	O
least	O
one	O
of	O
or	O
is	O
a	O
perfect	O
square	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
has	O
a	O
determinant	O
of	O
−1	O
,	O
and	O
thus	O
it	O
is	O
a	O
2×2	O
unimodular	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
occur	O
as	O
the	O
ratio	O
of	O
successive	O
convergents	O
of	O
the	O
continued	O
fraction	O
for	O
,	O
and	O
the	O
matrix	B-Architecture
formed	O
from	O
successive	O
convergents	O
of	O
any	O
continued	O
fraction	O
has	O
a	O
determinant	O
of	O
+1	O
or	O
−1	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
representation	O
gives	O
the	O
following	O
closed-form	O
expression	O
for	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
:	O
</s>
<s>
For	O
a	O
given	O
,	O
this	O
matrix	B-Architecture
can	O
be	O
computed	O
in	O
arithmetic	O
operations	O
,	O
using	O
the	O
exponentiation	B-Algorithm
by	I-Algorithm
squaring	I-Algorithm
method	O
.	O
</s>
<s>
Moreover	O
,	O
since	O
for	O
any	O
square	B-Algorithm
matrix	I-Algorithm
,	O
the	O
following	O
identities	O
can	O
be	O
derived	O
(	O
they	O
are	O
obtained	O
from	O
two	O
different	O
coefficients	O
of	O
the	O
matrix	B-Architecture
product	O
,	O
and	O
one	O
may	O
easily	O
deduce	O
the	O
second	O
one	O
from	O
the	O
first	O
one	O
by	O
changing	O
into	O
)	O
,	O
</s>
<s>
These	O
last	O
two	O
identities	O
provide	O
a	O
way	O
to	O
compute	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
recursively	O
in	O
arithmetic	O
operations	O
and	O
in	O
time	O
,	O
where	O
is	O
the	O
time	O
for	O
the	O
multiplication	O
of	O
two	O
numbers	O
of	O
digits	O
.	O
</s>
<s>
This	O
matches	O
the	O
time	O
for	O
computing	O
the	O
th	O
Fibonacci	B-Algorithm
number	I-Algorithm
from	O
the	O
closed-form	O
matrix	B-Architecture
formula	O
,	O
but	O
with	O
fewer	O
redundant	O
steps	O
if	O
one	O
avoids	O
recomputing	O
an	O
already	O
computed	O
Fibonacci	B-Algorithm
number	I-Algorithm
(	O
recursion	O
with	O
memoization	O
)	O
.	O
</s>
<s>
Most	O
identities	O
involving	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
can	O
be	O
proved	O
using	O
combinatorial	O
arguments	O
using	O
the	O
fact	O
that	O
can	O
be	O
interpreted	O
as	O
the	O
number	O
of	O
(	O
possibly	O
empty	O
)	O
sequences	O
of	O
1sand2s	O
whose	O
sum	O
is	O
.	O
</s>
<s>
In	O
the	O
following	O
,	O
is	O
the	O
cardinality	B-Application
of	O
a	O
set	O
:	O
</s>
<s>
Excluding	O
the	O
first	O
element	O
,	O
the	O
remaining	O
terms	O
in	O
each	O
sequence	O
sum	O
to	O
or	O
and	O
the	O
cardinality	B-Application
of	O
each	O
set	O
is	O
or	O
giving	O
a	O
total	O
of	O
sequences	O
,	O
showing	O
this	O
is	O
equal	O
to	O
.	O
</s>
<s>
In	O
a	O
similar	O
manner	O
it	O
may	O
be	O
shown	O
that	O
the	O
sum	O
of	O
the	O
first	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
up	O
to	O
the	O
nth	O
is	O
equal	O
to	O
the	O
(	O
n	O
+	O
2	O
)	O
nd	O
Fibonacci	B-Algorithm
number	I-Algorithm
minus1	O
.	O
</s>
<s>
Specifically	O
,	O
each	O
set	O
consists	O
of	O
those	O
sequences	O
that	O
start	O
until	O
the	O
last	O
two	O
sets	O
each	O
with	O
cardinality	B-Application
1	O
.	O
</s>
<s>
In	O
words	O
,	O
the	O
sum	O
of	O
the	O
first	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
with	O
odd	O
index	O
up	O
to	O
is	O
the	O
(	O
2n	O
)	O
th	O
Fibonacci	B-Algorithm
number	I-Algorithm
,	O
and	O
the	O
sum	O
of	O
the	O
first	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
with	O
even	O
index	O
up	O
to	O
is	O
the	O
(	O
2n+1	O
)	O
st	O
Fibonacci	B-Algorithm
number	I-Algorithm
minus1	O
.	O
</s>
<s>
or	O
in	O
words	O
,	O
the	O
sum	O
of	O
the	O
squares	O
of	O
the	O
first	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
up	O
to	O
is	O
the	O
product	O
of	O
the	O
nth	O
and	O
(	O
n+1	O
)	O
st	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
as	O
stated	O
above	O
,	O
the	O
-th	O
Fibonacci	B-Algorithm
number	I-Algorithm
equals	O
the	O
number	O
of	O
combinatorial	O
compositions	O
(	O
ordered	O
partitions	O
)	O
of	O
using	O
terms	O
1	O
and	O
2	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
ordinary	O
generating	O
function	O
of	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
,	O
i.e.	O
</s>
<s>
Fibonacci	O
identities	O
often	O
can	O
be	O
easily	O
proved	O
using	O
mathematical	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
where	O
Ln	O
is	O
the	O
n-th	O
Lucas	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
These	O
can	O
be	O
found	O
experimentally	O
using	O
lattice	O
reduction	O
,	O
and	O
are	O
useful	O
in	O
setting	O
up	O
the	O
special	B-Algorithm
number	I-Algorithm
field	I-Algorithm
sieve	I-Algorithm
to	O
factorize	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
Putting	O
in	O
this	O
formula	O
,	O
one	O
gets	O
again	O
the	O
formulas	O
of	O
the	O
end	O
of	O
above	O
section	O
Matrix	B-Architecture
form	O
.	O
</s>
<s>
Infinite	O
sums	O
over	O
reciprocal	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
can	O
sometimes	O
be	O
evaluated	O
in	O
terms	O
of	O
theta	O
functions	O
.	O
</s>
<s>
and	O
there	O
is	O
a	O
nested	O
sum	O
of	O
squared	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
giving	O
the	O
reciprocal	O
of	O
the	O
golden	O
ratio	O
,	O
</s>
<s>
Thus	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
is	O
an	O
example	O
of	O
a	O
divisibility	O
sequence	O
.	O
</s>
<s>
In	O
particular	O
,	O
any	O
three	O
consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
pairwise	O
coprime	O
because	O
both	O
and	O
.	O
</s>
<s>
Every	O
prime	O
number	O
p	O
divides	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
that	O
can	O
be	O
determined	O
by	O
the	O
value	O
of	O
p	O
modulo5	O
.	O
</s>
<s>
These	O
cases	O
can	O
be	O
combined	O
into	O
a	O
single	O
,	O
non-piecewise	O
formula	O
,	O
using	O
the	O
Legendre	O
symbol	O
:	O
</s>
<s>
When	O
m	O
is	O
largesay	O
a	O
500-bit	O
numberthen	O
we	O
can	O
calculate	O
Fm	O
(	O
mod	O
n	O
)	O
efficiently	O
using	O
the	O
matrix	B-Architecture
form	O
.	O
</s>
<s>
Here	O
the	O
matrix	B-Architecture
power	I-Architecture
Am	O
is	O
calculated	O
using	O
modular	O
exponentiation	O
,	O
which	O
can	O
be	O
adapted	O
to	O
matrices	O
.	O
</s>
<s>
A	O
Fibonacci	O
prime	O
is	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
that	O
is	O
prime	O
.	O
</s>
<s>
As	O
there	O
are	O
arbitrarily	O
long	O
runs	O
of	O
composite	O
numbers	O
,	O
there	O
are	O
therefore	O
also	O
arbitrarily	O
long	O
runs	O
of	O
composite	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
No	O
Fibonacci	B-Algorithm
number	I-Algorithm
greater	O
than	O
F6	O
=	O
8	O
is	O
one	O
greater	O
or	O
one	O
less	O
than	O
a	O
prime	O
number	O
.	O
</s>
<s>
The	O
only	O
nontrivial	O
square	O
Fibonacci	B-Algorithm
number	I-Algorithm
is	O
144	O
.	O
</s>
<s>
Attila	O
Pethő	O
proved	O
in	O
2001	O
that	O
there	O
is	O
only	O
a	O
finite	O
number	O
of	O
perfect	O
power	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
1	O
,	O
3	O
,	O
21	O
,	O
and	O
55	O
are	O
the	O
only	O
triangular	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
which	O
was	O
conjectured	O
by	O
Vern	O
Hoggatt	O
and	O
proved	O
by	O
Luo	O
Ming	O
.	O
</s>
<s>
No	O
Fibonacci	B-Algorithm
number	I-Algorithm
can	O
be	O
a	O
perfect	O
number	O
.	O
</s>
<s>
More	O
generally	O
,	O
no	O
Fibonacci	B-Algorithm
number	I-Algorithm
other	O
than	O
1	O
can	O
be	O
multiply	O
perfect	O
,	O
and	O
no	O
ratio	O
of	O
two	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
can	O
be	O
perfect	O
.	O
</s>
<s>
With	O
the	O
exceptions	O
of	O
1	O
,	O
8	O
and	O
144	O
(	O
F1	O
=	O
F2	O
,	O
F6	O
and	O
F12	O
)	O
every	O
Fibonacci	B-Algorithm
number	I-Algorithm
has	O
a	O
prime	O
factor	O
that	O
is	O
not	O
a	O
factor	O
of	O
any	O
smaller	O
Fibonacci	B-Algorithm
number	I-Algorithm
(	O
Carmichael	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
)	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
8	O
and	O
144	O
(	O
F6	O
and	O
F12	O
)	O
are	O
the	O
only	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
that	O
are	O
the	O
product	O
of	O
other	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
The	O
divisibility	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
by	O
a	O
prime	O
p	O
is	O
related	O
to	O
the	O
Legendre	O
symbol	O
which	O
is	O
evaluated	O
as	O
follows	O
:	O
</s>
<s>
All	O
known	O
factors	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
F(i )	O
for	O
all	O
i	O
<	O
50000	O
are	O
collected	O
at	O
the	O
relevant	O
repositories	O
.	O
</s>
<s>
If	O
the	O
members	O
of	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
are	O
taken	O
modn	O
,	O
the	O
resulting	O
sequence	O
is	O
periodic	O
with	O
period	O
at	O
most6n	O
.	O
</s>
<s>
The	O
lengths	O
of	O
the	O
periods	O
for	O
various	O
n	O
form	O
the	O
so-called	O
Pisano	B-Algorithm
periods	I-Algorithm
.	O
</s>
<s>
Determining	O
a	O
general	O
formula	O
for	O
the	O
Pisano	B-Algorithm
periods	I-Algorithm
is	O
an	O
open	O
problem	O
,	O
which	O
includes	O
as	O
a	O
subproblem	O
a	O
special	O
instance	O
of	O
the	O
problem	O
of	O
finding	O
the	O
multiplicative	O
order	O
of	O
a	O
modular	O
integer	O
or	O
of	O
an	O
element	O
in	O
a	O
finite	O
field	O
.	O
</s>
<s>
However	O
,	O
for	O
any	O
particular	O
n	O
,	O
the	O
Pisano	B-Algorithm
period	I-Algorithm
may	O
be	O
found	O
as	O
an	O
instance	O
of	O
cycle	B-Algorithm
detection	I-Algorithm
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
is	O
one	O
of	O
the	O
simplest	O
and	O
earliest	O
known	O
sequences	O
defined	O
by	O
a	O
recurrence	O
relation	O
,	O
and	O
specifically	O
by	O
a	O
linear	O
difference	O
equation	O
.	O
</s>
<s>
All	O
these	O
sequences	O
may	O
be	O
viewed	O
as	O
generalizations	O
of	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
.	O
</s>
<s>
Some	O
specific	O
examples	O
that	O
are	O
close	O
,	O
in	O
some	O
sense	O
,	O
to	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
include	O
:	O
</s>
<s>
Generalizing	O
the	O
index	O
to	O
negative	O
integers	O
to	O
produce	O
the	O
negafibonacci	B-Algorithm
numbers	O
.	O
</s>
<s>
Lucas	B-Algorithm
numbers	I-Algorithm
have	O
L1	O
=	O
1	O
,	O
L2	O
=	O
3	O
,	O
and	O
Ln	O
=	O
Ln−1	O
+	O
Ln−2	O
.	O
</s>
<s>
If	O
the	O
coefficient	O
of	O
the	O
preceding	O
value	O
is	O
assigned	O
a	O
variable	O
value	O
x	O
,	O
the	O
result	O
is	O
the	O
sequence	O
of	O
Fibonacci	B-Algorithm
polynomials	I-Algorithm
.	O
</s>
<s>
The	O
Padovan	B-Algorithm
sequence	I-Algorithm
and	O
Perrin	B-Algorithm
numbers	I-Algorithm
have	O
P(n )	O
=	O
P( n	O
−	O
2	O
)	O
+	O
P( n	O
−	O
3	O
)	O
.	O
</s>
<s>
The	O
resulting	O
sequences	O
are	O
known	O
as	O
n-Step	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
occur	O
in	O
the	O
sums	O
of	O
"	O
shallow	O
"	O
diagonals	O
in	O
Pascal	O
's	O
triangle	O
(	O
see	O
Binomial	O
coefficient	O
)	O
:	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
can	O
be	O
found	O
in	O
different	O
ways	O
among	O
the	O
set	O
of	O
binary	O
strings	O
,	O
or	O
equivalently	O
,	O
among	O
the	O
subsets	O
of	O
a	O
given	O
set	O
.	O
</s>
<s>
The	O
number	O
of	O
binary	O
strings	O
of	O
length	O
without	O
consecutive	O
s	O
is	O
the	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
A	O
bijection	B-Algorithm
with	O
the	O
sums	O
to	O
n+1	O
is	O
to	O
replace	O
1	O
with	O
0	O
and	O
2	O
with	O
10	O
,	O
and	O
drop	O
the	O
last	O
zero	O
.	O
</s>
<s>
The	O
number	O
of	O
binary	O
strings	O
of	O
length	O
without	O
an	O
odd	O
number	O
of	O
consecutive	O
s	O
is	O
the	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
A	O
bijection	B-Algorithm
with	O
the	O
sums	O
to	O
n	O
is	O
to	O
replace	O
1	O
with	O
0	O
and	O
2	O
with	O
11	O
.	O
</s>
<s>
Yuri	O
Matiyasevich	O
was	O
able	O
to	O
show	O
that	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
can	O
be	O
defined	O
by	O
a	O
Diophantine	O
equation	O
,	O
which	O
led	O
to	O
his	O
solving	O
Hilbert	O
's	O
tenth	O
problem	O
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
also	O
an	O
example	O
of	O
a	O
complete	O
sequence	O
.	O
</s>
<s>
This	O
means	O
that	O
every	O
positive	O
integer	O
can	O
be	O
written	O
as	O
a	O
sum	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
where	O
any	O
one	O
number	O
is	O
used	O
once	O
at	O
most	O
.	O
</s>
<s>
Moreover	O
,	O
every	O
positive	O
integer	O
can	O
be	O
written	O
in	O
a	O
unique	O
way	O
as	O
the	O
sum	O
of	O
one	O
or	O
more	O
distinct	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
in	O
such	O
a	O
way	O
that	O
the	O
sum	O
does	O
not	O
include	O
any	O
two	O
consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
This	O
is	O
known	O
as	O
Zeckendorf	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
,	O
and	O
a	O
sum	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
that	O
satisfies	O
these	O
conditions	O
is	O
called	O
a	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
The	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
of	O
a	O
number	O
can	O
be	O
used	O
to	O
derive	O
its	O
Fibonacci	B-Algorithm
coding	I-Algorithm
.	O
</s>
<s>
Starting	O
with	O
5	O
,	O
every	O
second	O
Fibonacci	B-Algorithm
number	I-Algorithm
is	O
the	O
length	O
of	O
the	O
hypotenuse	O
of	O
a	O
right	O
triangle	O
with	O
integer	O
sides	O
,	O
or	O
in	O
other	O
words	O
,	O
the	O
largest	O
number	O
in	O
a	O
Pythagorean	O
triple	O
,	O
obtained	O
from	O
the	O
formula	O
The	O
sequence	O
of	O
Pythagorean	O
triangles	O
obtained	O
from	O
this	O
formula	O
has	O
sides	O
of	O
lengths	O
(	O
3	O
,	O
4	O
,	O
5	O
)	O
,	O
(	O
5	O
,	O
12	O
,	O
13	O
)	O
,	O
(	O
16	O
,	O
30	O
,	O
34	O
)	O
,	O
(	O
39	O
,	O
80	O
,	O
89	O
)	O
,	O
....	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
cube	I-Algorithm
is	O
an	O
undirected	O
graph	O
with	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
of	O
nodes	O
that	O
has	O
been	O
proposed	O
as	O
a	O
network	B-Architecture
topology	I-Architecture
for	O
parallel	B-Operating_System
computing	I-Operating_System
.	O
</s>
<s>
Fibonacci	B-Algorithm
numbers	I-Algorithm
appear	O
in	O
the	O
ring	O
lemma	O
,	O
used	O
to	O
prove	O
connections	O
between	O
the	O
circle	O
packing	O
theorem	O
and	O
conformal	O
maps	O
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
important	O
in	O
computational	B-General_Concept
run-time	I-General_Concept
analysis	I-General_Concept
of	O
Euclid	O
's	O
algorithm	O
to	O
determine	O
the	O
greatest	O
common	O
divisor	O
of	O
two	O
integers	O
:	O
the	O
worst	O
case	O
input	O
for	O
this	O
algorithm	O
is	O
a	O
pair	O
of	O
consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
used	O
in	O
a	O
polyphase	O
version	O
of	O
the	O
merge	B-Algorithm
sort	I-Algorithm
algorithm	O
in	O
which	O
an	O
unsorted	O
list	O
is	O
divided	O
into	O
two	O
lists	O
whose	O
lengths	O
correspond	O
to	O
sequential	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
–	O
by	O
dividing	O
the	O
list	O
so	O
that	O
the	O
two	O
parts	O
have	O
lengths	O
in	O
the	O
approximate	O
proportion	O
φ	O
.	O
</s>
<s>
A	O
tape-drive	O
implementation	O
of	O
the	O
polyphase	B-Algorithm
merge	I-Algorithm
sort	I-Algorithm
was	O
described	O
in	O
The	B-General_Concept
Art	I-General_Concept
of	I-General_Concept
Computer	I-General_Concept
Programming	I-General_Concept
.	O
</s>
<s>
These	O
trees	O
have	O
a	O
number	O
of	O
vertices	O
that	O
is	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
minus	O
one	O
,	O
an	O
important	O
fact	O
in	O
the	O
analysis	O
of	O
AVL	O
trees	O
.	O
</s>
<s>
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
used	O
by	O
some	O
pseudorandom	B-Algorithm
number	I-Algorithm
generators	I-Algorithm
.	O
</s>
<s>
Fibonacci	B-Algorithm
numbers	I-Algorithm
arise	O
in	O
the	O
analysis	O
of	O
the	O
Fibonacci	B-Application
heap	I-Application
data	B-General_Concept
structure	I-General_Concept
.	O
</s>
<s>
A	O
one-dimensional	O
optimization	O
method	O
,	O
called	O
the	O
Fibonacci	B-Algorithm
search	I-Algorithm
technique	I-Algorithm
,	O
uses	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
number	I-Algorithm
series	O
is	O
used	O
for	O
optional	O
lossy	B-Algorithm
compression	I-Algorithm
in	O
the	O
IFF	B-Device
8SVX	B-Operating_System
audio	O
file	O
format	O
used	O
on	O
Amiga	B-Device
computers	I-Device
.	O
</s>
<s>
The	O
number	O
series	O
compands	B-Algorithm
the	O
original	O
audio	O
wave	O
similar	O
to	O
logarithmic	O
methods	O
such	O
as	O
μ-law	B-Algorithm
.	O
</s>
<s>
Some	O
Agile	O
teams	O
use	O
a	O
modified	O
series	O
called	O
the	O
"	O
Modified	O
Fibonacci	B-Algorithm
Series	I-Algorithm
"	O
in	O
planning	O
poker	O
,	O
as	O
an	O
estimation	O
tool	O
.	O
</s>
<s>
Fibonacci	B-Algorithm
sequences	I-Algorithm
appear	O
in	O
biological	O
settings	O
,	O
such	O
as	O
branching	O
in	O
trees	O
,	O
arrangement	O
of	O
leaves	O
on	O
a	O
stem	O
,	O
the	O
fruitlets	O
of	O
a	O
pineapple	O
,	O
the	O
flowering	O
of	O
artichoke	O
,	O
an	O
uncurling	O
fern	O
and	O
the	O
arrangement	O
of	O
a	O
pine	O
cone	O
,	O
and	O
the	O
family	O
tree	O
of	O
honeybees	O
.	O
</s>
<s>
Kepler	B-General_Concept
pointed	O
out	O
the	O
presence	O
of	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
in	O
nature	O
,	O
using	O
it	O
to	O
explain	O
the	O
(	O
golden	O
ratio-related	O
)	O
pentagonal	O
form	O
of	O
some	O
flowers	O
.	O
</s>
<s>
Field	O
daisies	O
most	O
often	O
have	O
petals	O
in	O
counts	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
In	O
1830	O
,	O
K	O
.	O
F	O
.	O
Schimper	O
and	O
A	O
.	O
Braun	O
discovered	O
that	O
the	O
parastichies	O
(	O
spiral	O
phyllotaxis	O
)	O
of	O
plants	O
were	O
frequently	O
expressed	O
as	O
fractions	O
involving	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
Przemysław	O
Prusinkiewicz	O
advanced	O
the	O
idea	O
that	O
real	O
instances	O
can	O
in	O
part	O
be	O
understood	O
as	O
the	O
expression	O
of	O
certain	O
algebraic	O
constraints	O
on	O
free	O
groups	O
,	O
specifically	O
as	O
certain	O
Lindenmayer	B-Language
grammars	I-Language
.	O
</s>
<s>
Sunflowers	O
and	O
similar	O
flowers	O
most	O
commonly	O
have	O
spirals	O
of	O
florets	O
in	O
clockwise	O
and	O
counter-clockwise	O
directions	O
in	O
the	O
amount	O
of	O
adjacent	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
typically	O
counted	O
by	O
the	O
outermost	O
range	O
of	O
radii	O
.	O
</s>
<s>
Fibonacci	B-Algorithm
numbers	I-Algorithm
also	O
appear	O
in	O
the	O
pedigrees	O
of	O
idealized	O
honeybees	O
,	O
according	O
to	O
the	O
following	O
rules	O
:	O
</s>
<s>
This	O
sequence	O
of	O
numbers	O
of	O
parents	O
is	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
.	O
</s>
<s>
It	O
has	O
been	O
noticed	O
that	O
the	O
number	O
of	O
possible	O
ancestors	O
on	O
the	O
human	O
X	O
chromosome	O
inheritance	O
line	O
at	O
a	O
given	O
ancestral	O
generation	O
also	O
follows	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
.	O
</s>
<s>
In	O
optics	B-Algorithm
,	O
when	O
a	O
beam	O
of	O
light	O
shines	O
at	O
an	O
angle	O
through	O
two	O
stacked	O
transparent	O
plates	O
of	O
different	O
materials	O
of	O
different	O
refractive	O
indexes	O
,	O
it	O
may	O
reflect	O
off	O
three	O
surfaces	O
:	O
the	O
top	O
,	O
middle	O
,	O
and	O
bottom	O
surfaces	O
of	O
the	O
two	O
plates	O
.	O
</s>
<s>
The	O
number	O
of	O
different	O
beam	O
paths	O
that	O
have	O
reflections	O
,	O
for	O
,	O
is	O
the	O
th	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
Since	O
the	O
conversion	O
factor	O
1.609344	O
for	O
miles	O
to	O
kilometers	O
is	O
close	O
to	O
the	O
golden	O
ratio	O
,	O
the	O
decomposition	O
of	O
distance	O
in	O
miles	O
into	O
a	O
sum	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
becomes	O
nearly	O
the	O
kilometer	O
sum	O
when	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
replaced	O
by	O
their	O
successors	O
.	O
</s>
<s>
This	O
method	O
amounts	O
to	O
a	O
radix	O
2	O
number	O
register	B-General_Concept
in	O
golden	O
ratio	O
base	O
φ	O
being	O
shifted	O
.	O
</s>
<s>
To	O
convert	O
from	O
kilometers	O
to	O
miles	O
,	O
shift	O
the	O
register	B-General_Concept
down	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
instead	O
.	O
</s>
<s>
The	O
measured	O
values	O
of	O
voltages	O
and	O
currents	O
in	O
the	O
infinite	O
resistor	O
chain	O
circuit	O
(	O
also	O
called	O
the	O
resistor	O
ladder	O
or	O
infinite	O
series-parallel	O
circuit	O
)	O
follow	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
.	O
</s>
<s>
The	O
intermediate	O
results	O
of	O
adding	O
the	O
alternating	O
series	O
and	O
parallel	O
resistances	O
yields	O
fractions	O
composed	O
of	O
consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
2012	O
show	O
how	O
a	O
generalized	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
also	O
can	O
be	O
connected	O
to	O
the	O
field	O
of	O
economics	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
is	O
shown	O
how	O
a	O
generalized	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
enters	O
the	O
control	O
function	O
of	O
finite-horizon	O
dynamic	O
optimisation	O
problems	O
with	O
one	O
state	O
and	O
one	O
control	O
variable	O
.	O
</s>
<s>
Mario	O
Merz	O
included	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
in	O
some	O
of	O
his	O
artworks	O
beginning	O
in	O
1970	O
.	O
</s>
