<s>
The	O
Lucas	B-Algorithm
sequence	I-Algorithm
is	O
an	O
integer	O
sequence	O
named	O
after	O
the	O
mathematician	O
François	O
Édouard	O
Anatole	O
Lucas	O
(	O
18421891	O
)	O
,	O
who	O
studied	O
both	O
that	O
sequence	O
and	O
the	O
closely	O
related	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
.	O
</s>
<s>
Individual	O
numbers	O
in	O
the	O
Lucas	B-Algorithm
sequence	I-Algorithm
are	O
known	O
as	O
Lucas	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
Lucas	B-Algorithm
numbers	I-Algorithm
and	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
form	O
complementary	O
instances	O
of	O
Lucas	B-Algorithm
sequences	I-Algorithm
.	O
</s>
<s>
The	O
Lucas	B-Algorithm
sequence	I-Algorithm
has	O
the	O
same	O
recursive	O
relationship	O
as	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
,	O
where	O
each	O
term	O
is	O
the	O
sum	O
of	O
the	O
two	O
previous	O
terms	O
,	O
but	O
with	O
different	O
starting	O
values	O
.	O
</s>
<s>
This	O
produces	O
a	O
sequence	O
where	O
the	O
ratios	O
of	O
successive	O
terms	O
approach	O
the	O
golden	O
ratio	O
,	O
and	O
in	O
fact	O
the	O
terms	O
themselves	O
are	O
roundings	B-Algorithm
of	O
integer	O
powers	O
of	O
the	O
golden	O
ratio	O
.	O
</s>
<s>
The	O
sequence	O
also	O
has	O
a	O
variety	O
of	O
relationships	O
with	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
like	O
the	O
fact	O
that	O
adding	O
any	O
two	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
two	O
terms	O
apart	O
in	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
results	O
in	O
the	O
Lucas	B-Algorithm
number	I-Algorithm
in	O
between	O
.	O
</s>
<s>
As	O
with	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
each	O
Lucas	B-Algorithm
number	I-Algorithm
is	O
defined	O
to	O
be	O
the	O
sum	O
of	O
its	O
two	O
immediately	O
previous	O
terms	O
,	O
thereby	O
forming	O
a	O
Fibonacci	O
integer	O
sequence	O
.	O
</s>
<s>
The	O
first	O
two	O
Lucas	B-Algorithm
numbers	I-Algorithm
are	O
and	O
,	O
which	O
differs	O
from	O
the	O
first	O
two	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
and	O
.	O
</s>
<s>
Though	O
closely	O
related	O
in	O
definition	O
,	O
Lucas	O
and	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
exhibit	O
distinct	O
properties	O
.	O
</s>
<s>
The	O
Lucas	B-Algorithm
numbers	I-Algorithm
may	O
thus	O
be	O
defined	O
as	O
follows	O
:	O
</s>
<s>
All	O
Fibonacci-like	O
integer	O
sequences	O
appear	O
in	O
shifted	O
form	O
as	O
a	O
row	O
of	O
the	O
Wythoff	B-Algorithm
array	I-Algorithm
;	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
itself	O
is	O
the	O
first	O
row	O
and	O
the	O
Lucas	B-Algorithm
sequence	I-Algorithm
is	O
the	O
second	O
row	O
.	O
</s>
<s>
Also	O
like	O
all	O
Fibonacci-like	O
integer	O
sequences	O
,	O
the	O
ratio	O
between	O
two	O
consecutive	O
Lucas	B-Algorithm
numbers	I-Algorithm
converges	B-Algorithm
to	O
the	O
golden	O
ratio	O
.	O
</s>
<s>
Using	O
,	O
one	O
can	O
extend	O
the	O
Lucas	B-Algorithm
numbers	I-Algorithm
to	O
negative	O
integers	O
to	O
obtain	O
a	O
doubly	O
infinite	O
sequence	O
:	O
</s>
<s>
The	O
Lucas	B-Algorithm
numbers	I-Algorithm
are	O
related	O
to	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
by	O
many	O
identities	O
.	O
</s>
<s>
Many	O
of	O
the	O
Fibonacci	O
identities	O
have	O
parallels	O
in	O
Lucas	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
be	O
the	O
generating	O
function	O
of	O
the	O
Lucas	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
then	O
no	O
Lucas	B-Algorithm
number	I-Algorithm
is	O
divisible	O
by	O
.	O
</s>
<s>
A	O
Lucas	O
prime	O
is	O
a	O
Lucas	B-Algorithm
number	I-Algorithm
that	O
is	O
prime	O
.	O
</s>
<s>
,	O
the	O
largest	O
known	O
Lucas	O
probable	B-Algorithm
prime	I-Algorithm
is	O
L5466311	O
,	O
with	O
1,142,392	O
decimal	O
digits	O
.	O
</s>
<s>
In	O
the	O
same	O
way	O
as	O
Fibonacci	B-Algorithm
polynomials	I-Algorithm
are	O
derived	O
from	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
the	O
Lucas	B-Algorithm
polynomials	I-Algorithm
are	O
a	O
polynomial	O
sequence	O
derived	O
from	O
the	O
Lucas	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
Lucas	B-Algorithm
numbers	I-Algorithm
are	O
the	O
second	O
most	O
common	O
pattern	O
in	O
sunflowers	O
after	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
when	O
clockwise	O
and	O
counter-clockwise	O
spirals	O
are	O
counted	O
,	O
according	O
to	O
an	O
analysis	O
of	O
657	O
sunflowers	O
in	O
2016	O
.	O
</s>
