Program for Perrin numbers

The Perrin numbers are the numbers in the following integer sequence.
3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39 …

In mathematical terms, the sequence p(n) of Perrin numbers is defined by the recurrence relation

 P(n) = P(n-2) + P(n-3) for n > 2, 

with initial values
    P(0) = 3, P(1) = 0, P(2) = 2. 

Write a function int per(int n) that returns p(n). For example, if n = 0, then per() should return 3. If n = 1, then it should return 0 If n = 2, then it should return 2. For n > 2, it should return p(n-2) + p(n-3)



Method 1 ( Use recursion : Exponential )
Below is simple recursive implementation of above formula.

C++

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// n'th perrin number using Recursion'
#include <bits/stdc++.h>
using namespace std;
  
int per(int n)
{
    if (n == 0)
        return 3;
    if (n == 1)
        return 0;
    if (n == 2)
        return 2;
    return per(n - 2) + per(n - 3);
}
  
// Driver code
int main()
{
    int n = 9;
    cout << per(n);
    return 0;
}
  
// This code is contributed 
// by Akanksha Rai

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C

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// n'th perrin number using Recursion'
#include <stdio.h>
int per(int n)
{
    if (n == 0)
        return 3;
    if (n == 1)
        return 0;
    if (n == 2)
        return 2;
    return per(n - 2) + per(n - 3);
}
  
// Driver code
int main()
{
    int n = 9;
    printf("%d", per(n));
    return 0;
}

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Java

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// Java code for n'th perrin number
// using Recursion'
import java.io.*;
  
class GFG {
  
    static int per(int n)
    {
        if (n == 0)
            return 3;
        if (n == 1)
            return 0;
        if (n == 2)
            return 2;
        return per(n - 2) + per(n - 3);
    }
  
    // Driver code
    public static void main(String[] args)
    {
  
        int n = 9;
  
        System.out.println(per(n));
    }
}
  
// This code is contributed by vt_m.

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Python3

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# Python3 code for n'th perrin 
# number using Recursion'
  
# function return n'th
# perrin number
def per(n):
  
    if (n == 0):
        return 3;
    if (n == 1):
        return 0;
    if (n == 2):
        return 2;
    return per(n - 2) + per(n - 3);
  
# Driver Code
n = 9;
print(per(n));
      
# This code is contributed mits

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C#

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// C# code for n'th perrin number
// using Recursion'
using System;
  
class GFG {
  
    static int per(int n)
    {
        if (n == 0)
            return 3;
        if (n == 1)
            return 0;
        if (n == 2)
            return 2;
        return per(n - 2) + per(n - 3);
    }
  
    // Driver code
    public static void Main()
    {
  
        int n = 9;
  
        Console.Write(per(n));
    }
}
  
// This code is contributed by vt_m.

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PHP

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<?php
// PHP code for n'th perrin 
// number using Recursion'
  
// function return n'th
// perrin number
function per($n)
{
    if ($n == 0)
        return 3;
    if ($n == 1)
        return 0;
    if ($n == 2)
        return 2;
    return per($n - 2) + 
           per($n - 3);
}
  
    // Driver Code
    $n = 9;
    echo per($n);
      
#This code is contributed ajit.
?>

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Output:

12

We see that in this implementation a lot of repeated work in the following recursion tree.

                           per(8)   
                       /           \     
               per(6)             per(5)   
              /      \             /     \
        per(4)      per(3)        per(3)    per(2)
       /     \        /    \        /  \  
   per(2)   per(1)  per(1) per(0) per(1) per(0)

Method 2: ( Optimized : Linear)

C++



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// Optimized C++ program for n'th perrin number
#include <bits/stdc++.h>
using namespace std;
int per(int n)
{
    int a = 3, b = 0, c = 2, i;
    int m;
    if (n == 0)
        return a;
    if (n == 1)
        return b;
    if (n == 2)
        return c;
    while (n > 2) {
        m = a + b;
        a = b;
        b = c;
        c = m;
        n--;
    }
    return m;
}
  
// Driver code
int main()
{
    int n = 9;
    cout << per(n);
    return 0;
}
  
// This code is contributed 
// by Akanksha Rai

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C

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// Optimized C program for n'th perrin number
#include <stdio.h>
int per(int n)
{
    int a = 3, b = 0, c = 2, i;
    int m;
    if (n == 0)
        return a;
    if (n == 1)
        return b;
    if (n == 2)
        return c;
    while (n > 2) {
        m = a + b;
        a = b;
        b = c;
        c = m;
        n--;
    }
    return m;
}
  
// Driver code
int main()
{
    int n = 9;
    printf("%d", per(n));
    return 0;
}

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Java

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// Optimized Java program for n'th perrin number
import java.io.*;
  
class GFG {
  
    static int per(int n)
    {
        int a = 3, b = 0, c = 2, i;
        int m = 0;
        if (n == 0)
            return a;
        if (n == 1)
            return b;
        if (n == 2)
            return c;
        while (n > 2) {
            m = a + b;
            a = b;
            b = c;
            c = m;
            n--;
        }
        return m;
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int n = 9;
  
        System.out.println(per(n));
    }
}
  
// This code is contributed by vt_m.

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C#

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// Optimized C# program for n'th perrin number
using System;
  
class GFG {
  
    static int per(int n)
    {
        int a = 3, b = 0, c = 2;
  
        // int i;
        int m = 0;
        if (n == 0)
            return a;
        if (n == 1)
            return b;
        if (n == 2)
            return c;
  
        while (n > 2) {
            m = a + b;
            a = b;
            b = c;
            c = m;
            n--;
        }
  
        return m;
    }
  
    // Driver code
    public static void Main()
    {
  
        int n = 9;
  
        Console.WriteLine(per(n));
    }
}
  
// This code is contributed by vt_m.

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PHP

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<?php
// Optimized PHP program for 
// n'th perrin number
  
// function return the 
// n'th perrin number
function per($n)
{
    $a = 3; $b = 0; 
    $c = 2; $i;
    $m;
    if ($n == 0)
        return $a;
    if ($n == 1)
        return $b;
    if ($n == 2)
        return $c;
    while ($n > 2) 
    {
        $m = $a + $b;
        $a = $b;
        $b = $c;
        $c = $m;
        $n--;
    }
    return $m;
}
  
    // Driver code
    $n = 9;
    echo per($n);
      
// This code is contributed by ajit
?>

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Output:

12

Time Complexity : O(n)
Auxiliary Space : O(1)

Method 3: (Further Optimized : Logarithmic)
We can further optimize using Matrix Exponentiation. The matrix power formula for n’th Perrin number is

{\Huge \begin{pmatrix}  0& 1 & 0\\   0&  0&1 \\  1 &1  & 0 \\ \end{pmatrix}^n \begin{pmatrix} 3\\  0\\  2 \end{pmatrix} = \begin{pmatrix} P(n)\\  P(n+1)\\  P(n+2) \end{pmatrix}}

We can implement this method similar to implementation of method 5 of Fibonacci numbers. Since we can compute n’th power of a constant matrix in O(Log n), time complexity of this method is O(Log n)

Application :
The number of different maximal independent sets in an n-vertex cycle graph is counted by the nth Perrin number for n > 1

Related Article :
Sum of Perrin Numbers

Reference:
https://en.wikipedia.org/wiki/Perrin_number

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