<s>
In	O
mathematics	O
,	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
form	O
a	O
sequence	O
defined	O
recursively	O
by	O
:	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
has	O
been	O
studied	O
extensively	O
and	O
generalized	O
in	O
many	O
ways	O
,	O
for	O
example	O
,	O
by	O
starting	O
with	O
other	O
numbers	O
than	O
0	O
and	O
1	O
,	O
by	O
adding	O
more	O
than	O
two	O
numbers	O
to	O
generate	O
the	O
next	O
number	O
,	O
or	O
by	O
adding	O
objects	O
other	O
than	O
numbers	O
.	O
</s>
<s>
Using	O
,	O
one	O
can	O
extend	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
to	O
negative	O
integers	O
.	O
</s>
<s>
See	O
also	O
Negafibonacci	B-Algorithm
coding	I-Algorithm
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
possible	O
generalizations	O
of	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
which	O
include	O
the	O
real	O
numbers	O
(	O
and	O
sometimes	O
the	O
complex	O
numbers	O
)	O
in	O
their	O
domain	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
analytic	B-Language
function	I-Language
:	O
</s>
<s>
Since	O
for	O
all	O
complex	O
numbers	O
,	O
this	O
function	O
also	O
provides	O
an	O
extension	O
of	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
to	O
the	O
entire	O
complex	O
plane	O
.	O
</s>
<s>
Hence	O
we	O
can	O
calculate	O
the	O
generalized	O
Fibonacci	B-Algorithm
function	I-Algorithm
of	O
a	O
complex	O
variable	O
,	O
for	O
example	O
,	O
</s>
<s>
The	O
term	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
is	O
also	O
applied	O
more	O
generally	O
to	O
any	O
function	O
from	O
the	O
integers	O
to	O
a	O
field	O
for	O
which	O
.	O
</s>
<s>
These	O
functions	O
are	O
precisely	O
those	O
of	O
the	O
form	O
,	O
so	O
the	O
Fibonacci	B-Algorithm
sequences	I-Algorithm
form	O
a	O
vector	O
space	O
with	O
the	O
functions	O
and	O
as	O
a	O
basis	O
.	O
</s>
<s>
Then	O
the	O
Fibonacci	B-Algorithm
sequences	I-Algorithm
form	O
a	O
2-dimensional	O
Z-module	O
in	O
the	O
same	O
way	O
.	O
</s>
<s>
The	O
ratio	O
between	O
two	O
consecutive	O
elements	O
converges	B-Algorithm
to	O
the	O
golden	O
ratio	O
,	O
except	O
in	O
the	O
case	O
of	O
the	O
sequence	O
which	O
is	O
constantly	O
zero	O
and	O
the	O
sequences	O
where	O
the	O
ratio	O
of	O
the	O
two	O
first	O
terms	O
is	O
.	O
</s>
<s>
In	O
this	O
form	O
the	O
simplest	O
non-trivial	O
example	O
has	O
,	O
which	O
is	O
the	O
sequence	O
of	O
Lucas	B-Algorithm
numbers	I-Algorithm
:	O
</s>
<s>
Every	O
nontrivial	O
Fibonacci	O
integer	O
sequence	O
appears	O
(	O
possibly	O
after	O
a	O
shift	O
by	O
a	O
finite	O
number	O
of	O
positions	O
)	O
as	O
one	O
of	O
the	O
rows	O
of	O
the	O
Wythoff	B-Algorithm
array	I-Algorithm
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
itself	O
is	O
the	O
first	O
row	O
,	O
and	O
a	O
shift	O
of	O
the	O
Lucas	B-Algorithm
sequence	I-Algorithm
is	O
the	O
second	O
row	O
.	O
</s>
<s>
A	O
different	O
generalization	O
of	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
is	O
the	O
Lucas	B-Algorithm
sequences	I-Algorithm
of	O
the	O
kind	O
defined	O
as	O
follows	O
:	O
</s>
<s>
where	O
the	O
normal	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
is	O
the	O
special	O
case	O
of	O
and	O
.	O
</s>
<s>
Another	O
kind	O
of	O
Lucas	B-Algorithm
sequence	I-Algorithm
begins	O
with	O
,	O
.	O
</s>
<s>
A	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
of	O
order	O
is	O
an	O
integer	O
sequence	O
in	O
which	O
each	O
sequence	O
element	O
is	O
the	O
sum	O
of	O
the	O
previous	O
elements	O
(	O
with	O
the	O
exception	O
of	O
the	O
first	O
elements	O
in	O
the	O
sequence	O
)	O
.	O
</s>
<s>
The	O
usual	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
a	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
of	O
order	O
2	O
.	O
</s>
<s>
The	O
number	O
of	O
compositions	O
of	O
nonnegative	O
integers	O
into	O
parts	O
that	O
are	O
at	O
most	O
is	O
a	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
of	O
order	O
.	O
</s>
<s>
The	O
sequence	O
of	O
the	O
number	O
of	O
strings	O
of	O
0s	O
and	O
1s	O
of	O
length	O
that	O
contain	O
at	O
most	O
consecutive	O
0s	O
is	O
also	O
a	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
of	O
order	O
.	O
</s>
<s>
The	O
tribonacci	O
numbers	O
are	O
like	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
but	O
instead	O
of	O
starting	O
with	O
two	O
predetermined	O
terms	O
,	O
the	O
sequence	O
starts	O
with	O
three	O
predetermined	O
terms	O
and	O
each	O
term	O
afterwards	O
is	O
the	O
sum	O
of	O
the	O
preceding	O
three	O
terms	O
.	O
</s>
<s>
The	O
tetranacci	B-Algorithm
constant	I-Algorithm
is	O
the	O
ratio	O
toward	O
which	O
adjacent	O
tetranacci	O
numbers	O
tend	O
.	O
</s>
<s>
The	O
tetranacci	B-Algorithm
constant	I-Algorithm
can	O
be	O
expressed	O
in	O
terms	O
of	O
radicals	O
by	O
the	O
following	O
expression	O
:	O
</s>
<s>
The	O
special	O
case	O
is	O
the	O
traditional	O
Fibonacci	B-Algorithm
series	I-Algorithm
yielding	O
the	O
golden	O
section	O
.	O
</s>
<s>
where	O
denotes	O
the	O
nearest	B-Algorithm
integer	I-Algorithm
function	I-Algorithm
and	O
is	O
the	O
-nacci	O
constant	O
,	O
which	O
is	O
the	O
root	O
of	O
nearest	O
to	O
2	O
.	O
</s>
<s>
In	O
analogy	O
to	O
its	O
numerical	O
counterpart	O
,	O
the	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
defined	O
by	O
:	O
</s>
<s>
The	O
sequence	O
of	O
Fibonacci	B-Algorithm
strings	I-Algorithm
starts	O
:	O
</s>
<s>
The	O
length	O
of	O
each	O
Fibonacci	B-Algorithm
string	I-Algorithm
is	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
,	O
and	O
similarly	O
there	O
exists	O
a	O
corresponding	O
Fibonacci	B-Algorithm
string	I-Algorithm
for	O
each	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
Fibonacci	B-Algorithm
strings	I-Algorithm
appear	O
as	O
inputs	O
for	O
the	O
worst	B-General_Concept
case	I-General_Concept
in	O
some	O
computer	O
algorithms	O
.	O
</s>
<s>
If	O
"	O
a	O
"	O
and	O
"	O
b	O
"	O
represent	O
two	O
different	O
materials	O
or	O
atomic	O
bond	O
lengths	O
,	O
the	O
structure	O
corresponding	O
to	O
a	O
Fibonacci	B-Algorithm
string	I-Algorithm
is	O
a	O
Fibonacci	B-Algorithm
quasicrystal	I-Algorithm
,	O
an	O
aperiodic	O
quasicrystal	O
structure	O
with	O
unusual	O
spectral	O
properties	O
.	O
</s>
<s>
A	O
convolved	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
is	O
obtained	O
applying	O
a	O
convolution	B-Language
operation	O
to	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
one	O
or	O
more	O
times	O
.	O
</s>
<s>
where	O
is	O
the	O
th	O
derivative	B-Algorithm
of	O
.	O
</s>
<s>
The	O
numbers	O
are	O
the	O
row	O
sums	O
of	O
Hosoya	B-Algorithm
's	I-Algorithm
triangle	I-Algorithm
.	O
</s>
<s>
As	O
with	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
there	O
are	O
several	O
combinatorial	O
interpretations	O
of	O
these	O
sequences	O
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
polynomials	I-Algorithm
are	O
another	O
generalization	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
The	O
Padovan	B-Algorithm
sequence	I-Algorithm
is	O
generated	O
by	O
the	O
recurrence	O
.	O
</s>
<s>
A	O
random	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
can	O
be	O
defined	O
by	O
tossing	O
a	O
coin	O
for	O
each	O
position	O
of	O
the	O
sequence	O
and	O
taking	O
if	O
it	O
lands	O
heads	O
and	O
if	O
it	O
lands	O
tails	O
.	O
</s>
<s>
A	O
repfigit	B-Algorithm
,	O
or	O
Keith	B-Algorithm
number	I-Algorithm
,	O
is	O
an	O
integer	O
such	O
that	O
,	O
when	O
its	O
digits	O
start	O
a	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
with	O
that	O
number	O
of	O
digits	O
,	O
the	O
original	O
number	O
is	O
eventually	O
reached	O
.	O
</s>
<s>
An	O
example	O
is	O
47	O
,	O
because	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
starting	O
with	O
4	O
and	O
7	O
reaches	O
47	O
.	O
</s>
<s>
A	O
repfigit	B-Algorithm
can	O
be	O
a	O
tribonacci	O
sequence	O
if	O
there	O
are	O
3	O
digits	O
in	O
the	O
number	O
,	O
a	O
tetranacci	O
number	O
if	O
the	O
number	O
has	O
four	O
digits	O
,	O
etc	O
.	O
</s>
<s>
The	O
first	O
few	O
repfigits	B-Algorithm
are	O
:	O
</s>
<s>
If	O
we	O
abbreviate	O
such	O
a	O
sequence	O
as	O
,	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
and	O
the	O
shifted	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
are	O
seen	O
to	O
form	O
a	O
canonical	O
basis	O
for	O
this	O
space	O
,	O
yielding	O
the	O
identity	O
:	O
</s>
