<s>
The	O
Leonardo	B-Algorithm
numbers	I-Algorithm
are	O
a	O
sequence	O
of	O
numbers	O
given	O
by	O
the	O
recurrence	O
:	O
</s>
<s>
Edsger	O
W	O
.	O
Dijkstra	O
used	O
them	O
as	O
an	O
integral	O
part	O
of	O
his	O
smoothsort	B-Application
algorithm	O
,	O
and	O
also	O
analyzed	O
them	O
in	O
some	O
detail	O
.	O
</s>
<s>
A	O
Leonardo	B-Algorithm
prime	I-Algorithm
is	O
a	O
Leonardo	B-Algorithm
number	I-Algorithm
that	O
's	O
also	O
prime	O
.	O
</s>
<s>
1	O
,	O
1	O
,	O
3	O
,	O
5	B-Application
,	O
9	O
,	O
15	O
,	O
25	O
,	O
41	O
,	O
67	O
,	O
109	O
,	O
177	O
,	O
287	O
,	O
465	O
,	O
753	O
,	O
1219	O
,	O
1973	O
,	O
3193	O
,	O
5167	O
,	O
8361	O
,	O
...	O
</s>
<s>
3	O
,	O
5	B-Application
,	O
41	O
,	O
67	O
,	O
109	O
,	O
1973	O
,	O
5167	O
,	O
2692537	O
,	O
11405773	O
,	O
126491971	O
,	O
331160281	O
,	O
535828591	O
,	O
279167724889	O
,	O
145446920496281	O
,	O
28944668049352441	O
,	O
5760134388741632239	O
,	O
63880869269980199809	O
,	O
167242286979696845953	O
,	O
597222253637954133837103	O
,	O
...	O
</s>
<s>
The	O
Leonardo	B-Algorithm
numbers	I-Algorithm
form	O
a	O
cycle	O
in	O
any	O
modulo	O
n≥2	O
.	O
</s>
<s>
The	O
Leonardo	B-Algorithm
numbers	I-Algorithm
are	O
related	O
to	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
by	O
the	O
relation	O
.	O
</s>
<s>
From	O
this	O
relation	O
it	O
is	O
straightforward	O
to	O
derive	O
a	O
closed-form	O
expression	O
for	O
the	O
Leonardo	B-Algorithm
numbers	I-Algorithm
,	O
analogous	O
to	O
Binet	O
's	O
formula	O
for	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
:	O
</s>
