<s>
A	O
Padovan	B-Algorithm
prime	I-Algorithm
is	O
a	O
Padovan	B-Algorithm
number	I-Algorithm
that	O
is	O
prime	O
.	O
</s>
<s>
The	O
first	O
Padovan	B-Algorithm
primes	I-Algorithm
are	O
:	O
</s>
<s>
The	O
Padovan	B-Algorithm
sequence	I-Algorithm
is	O
named	O
after	O
Richard	O
Padovan	O
who	O
attributed	O
its	O
discovery	O
to	O
Dutch	O
architect	O
Hans	O
van	O
der	O
Laan	O
in	O
his	O
1994	O
essay	O
Dom	O
.	O
</s>
<s>
The	O
Perrin	B-Algorithm
sequence	I-Algorithm
satisfies	O
the	O
same	O
recurrence	O
relations	O
as	O
the	O
Padovan	B-Algorithm
sequence	I-Algorithm
,	O
although	O
it	O
has	O
different	O
initial	O
values	O
.	O
</s>
<s>
The	O
sum	O
of	O
the	O
first	O
n	O
terms	O
in	O
the	O
Padovan	B-Algorithm
sequence	I-Algorithm
is	O
2	O
less	O
than	O
P( n+5	O
)	O
,	O
i.e.	O
</s>
<s>
Sums	O
involving	O
products	O
of	O
terms	O
in	O
the	O
Padovan	B-Algorithm
sequence	I-Algorithm
satisfy	O
the	O
following	O
identities	O
:	O
</s>
<s>
The	O
Padovan	B-Algorithm
sequence	I-Algorithm
is	O
related	O
to	O
sums	O
of	O
binomial	O
coefficients	O
by	O
the	O
following	O
identity	O
:	O
</s>
<s>
This	O
equation	O
has	O
3	O
roots	O
;	O
one	O
real	O
root	O
p	O
(	O
known	O
as	O
the	O
plastic	O
number	O
)	O
and	O
two	O
complex	O
conjugate	O
roots	O
q	O
and	O
r	O
.	O
Given	O
these	O
three	O
roots	O
,	O
the	O
Padovan	B-Algorithm
sequence	I-Algorithm
can	O
be	O
expressed	O
by	O
a	O
formula	O
involving	O
p	O
,	O
q	O
and	O
r	O
:	O
</s>
<s>
Since	O
the	O
absolute	O
values	O
of	O
the	O
complex	O
roots	O
q	O
and	O
r	O
are	O
both	O
less	O
than	O
1	O
(	O
and	O
hence	O
p	O
is	O
a	O
Pisot	O
–	O
Vijayaraghavan	O
number	O
)	O
,	O
the	O
powers	O
of	O
these	O
roots	O
approach	B-Algorithm
0	O
for	O
large	O
n	O
,	O
and	O
tends	O
to	O
zero	O
.	O
</s>
<s>
For	O
all	O
,	O
P(n )	O
is	O
the	O
integer	B-Algorithm
closest	I-Algorithm
to	O
.	O
</s>
<s>
The	O
ratio	O
of	O
successive	O
terms	O
in	O
the	O
Padovan	B-Algorithm
sequence	I-Algorithm
approaches	O
p	O
,	O
which	O
has	O
a	O
value	O
of	O
approximately	O
1.324718	O
.	O
</s>
<s>
This	O
constant	O
bears	O
the	O
same	O
relationship	O
to	O
the	O
Padovan	B-Algorithm
sequence	I-Algorithm
and	O
the	O
Perrin	B-Algorithm
sequence	I-Algorithm
as	O
the	O
golden	O
ratio	O
does	O
to	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
used	O
to	O
prove	O
identities	O
involving	O
products	O
of	O
the	O
Padovan	B-Algorithm
sequence	I-Algorithm
with	O
geometric	O
terms	O
,	O
such	O
as	O
:	O
</s>
<s>
then	O
this	O
Lindenmayer	B-Language
system	I-Language
or	O
L-system	B-Language
produces	O
the	O
following	O
sequence	O
of	O
strings	O
:	O
</s>
<s>
and	O
if	O
we	O
count	O
the	O
length	O
of	O
each	O
string	O
,	O
we	O
obtain	O
the	O
Padovan	B-Algorithm
numbers	I-Algorithm
:	O
</s>
<s>
and	O
CC	O
pairs	O
are	O
also	O
Padovan	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
A	O
spiral	O
can	O
be	O
formed	O
based	O
on	O
connecting	O
the	O
corners	O
of	O
a	O
set	O
of	O
3-dimensional	O
cuboids	B-General_Concept
.	O
</s>
<s>
This	O
is	O
the	O
Padovan	O
cuboid	B-General_Concept
spiral	O
.	O
</s>
<s>
the	O
Padovan	B-Algorithm
numbers	I-Algorithm
multiplied	O
by	O
the	O
square	O
root	O
of	O
2	O
.	O
</s>
<s>
The	O
Padovan	B-Algorithm
numbers	I-Algorithm
were	O
discovered	O
in	O
1994	O
.	O
</s>
<s>
Paul	O
Barry	O
(	O
2004	O
)	O
showed	O
that	O
these	O
diagonals	O
generate	O
the	O
Padovan	B-Algorithm
sequence	I-Algorithm
by	O
summing	O
the	O
diagonal	O
numbers	O
.	O
</s>
