Padovan Sequence

Last Updated : 24 Jan, 2022

Padovan Sequence similar to Fibonacci sequence with similar recursive structure. The recursive formula is,

  P(n) = P(n-2) + P(n-3)
  P(0) = P(1) = P(2) = 1

Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55......
Spiral of squares with side lengths which follow the Fibonacci sequence.

Padovan Sequence: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37,.....
Spiral of equilateral triangles with side lengths which follow the Padovan sequence.

Examples:

For Padovan Sequence:
P0 = P1 = P2 = 1 ,
P(7) = P(5) + P(4)
     = P(3) + P(2) + P(2) + P(1)
     = P(2) + P(1) + 1 + 1 + 1
     = 1 + 1 + 1 + 1 + 1 
     = 5

C++

// C++ program to find n'th term in Padovan Sequence
// using Dynamic Programming
#include<iostream>
using namespace std;

/* Function to calculate padovan number P(n) */
int pad(int n)
{
    /* 0th ,1st and 2nd number of the series are 1*/
    int pPrevPrev = 1, pPrev = 1, pCurr = 1, pNext = 1;

    for (int i=3; i<=n; i++)
    {
        pNext = pPrevPrev + pPrev;
        pPrevPrev = pPrev;
        pPrev = pCurr;
        pCurr = pNext;
    }

    return pNext;
}

/* Driver Program */
int main()
{
    int n = 12;
    cout << pad(n);
    return 0;
}

Java

// Java program to find n'th term
// in Padovan Sequence using
// Dynamic Programming
import java.io.*;

class GFG {
    
    /* Function to calculate
    padovan number P(n) */
    static int pad(int n)
    {
       int []padv=new int[n]; //create array to store padovan values
       padv[0]=padv[1]=padv[2]=1;
        for (int i = 3; i <= n; i++) {
         padv[i]=padv[i-2]+padv[i-3];   
        }
        return padv[n-1];

        
    }

    /* Driver Program */
    public static void main(String args[])
    {
        int n = 12;
        System.out.println(pad(n));
    }
}

/*This code is contributed by Kanjam Bhat Lidhoo.*/

Python3

# Python program to find n'th term in Padovan
# Sequence using Dynamic Programming

# Function to calculate padovan number P(n)
def pad(n):

    # 0th ,1st and 2nd number of the series are 1
    pPrevPrev, pPrev, pCurr, pNext = 1, 1, 1, 1

    # Find n'th Padovan number using recursive
    # formula.
    for i in range(3, n+1):
        pNext = pPrevPrev + pPrev
        pPrevPrev = pPrev
        pPrev = pCurr
        pCurr = pNext

    return pNext

# Driver Code
print (pad(12))

// C# program to find n'th term
// in Padovan Sequence using
// Dynamic Programming
using System;

class GFG {

    /* Function to calculate
    padovan number P(n) */
    static int pad(int n)
    {
        
        /* 0th, 1st and 2nd number 
        of the series are 1*/
        int pPrevPrev = 1, pPrev = 1,
            pCurr = 1, pNext = 1;

        for (int i = 3; i <= n; i++) {
            pNext = pPrevPrev + pPrev;
            pPrevPrev = pPrev;
            pPrev = pCurr;
            pCurr = pNext;
        }

        return pNext;
    }

    /* Driver Program */
    public static void Main()
    {
        int n = 12;
        
        Console.WriteLine(pad(n));
    }
}

/*This code is contributed by vt_m.*/

PHP

<?php
// PHP program to find n'th 
// term in Padovan Sequence
// using Dynamic Programming

// Function to calculate 
// padovan number P(n)
function pad($n)
{
    
    // 0th ,1st and 2nd number
    // of the series are 1
    $pPrevPrev = 1; $pPrev = 1; 
    $pCurr = 1; $pNext = 1;

    for ($i = 3; $i <= $n; $i++)
    {
        $pNext = $pPrevPrev + $pPrev;
        $pPrevPrev = $pPrev;
        $pPrev = $pCurr;
        $pCurr = $pNext;
    }

    return $pNext;
}

// Driver Code
$n = 12;
echo(pad($n));

// This code is contributed by Ajit.
?>

JavaScript

<script>
// Javascript program to find n'th
// term in Padovan Sequence
// using Dynamic Programming

// Function to calculate
// padovan number P(n)
function pad(n) {

    // 0th ,1st and 2nd number
    // of the series are 1
    let pPrevPrev = 1;
    let pPrev = 1;
    let pCurr = 1;
    let pNext = 1;

    for (let i = 3; i <= n; i++) {
        pNext = pPrevPrev + pPrev;
        pPrevPrev = pPrev;
        pPrev = pCurr;
        pCurr = pNext;
    }

    return pNext;
}

// Driver Code
let n = 12;
document.write(pad(n));

// This code is contributed by gfgking.
</script>

Output:

Count of different ways to express N as the sum of 1, 3 and 4

kartik

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