The Padovan sequence

Date: 6-jun-2010

The Padovan Saquence is a set of integers that has interesting properties. To construct this sequece we are going to initialize the three first item in our vector to 1.
So,

P[0]=1;
P[1]=1;
P[2]=1;

And then apply the equation:
P[i]=P[i-2] + P[i-3]

Here is the code:

/**
 * Simple script to demonstrate the Padovan sequence
 * having the equation P[i]=P[i-2] + P[i-3]
 * where P[0]=P[1]=P[2]=1
 */

 /* --- Initialize the three first values to 1 in the vector */
 $P[0]=1;
 $P[1]=1;
 $P[2]=1;
 
 $i=3;
 $numResults=21;
 do
 {
    /* --- Apply equation P[i]=P[i-2] + P[i-3] */
    $P[$i]=$P[$i-2] + $P[$i-3];
    $i++;
 }
 while ($i<$numResults); 
 

 /* --- Print values */
 for($i=0;$i<=$numResults;$i++)
 {
    echo "\n".$P[$i];
 }
 
 
/*
Will print:
1
1
1
2
2
3
4
5
7
9
12
16
21
28
37
49
65
86
114
151
200
*/

Now look at this image, it is a spiral that follow the Padovan sequence:

This image proofs the property:

P(n) = P(n − 1) + P(n − 5)

You can see it clearly the triangle sized 9 is equel to the addition of the two side triangles (7 + 2)


You can check all this in Wiki