Padovan Sequence

1

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.
fibonacci-tiles1
Image Source : https://en.wikipedia.org/wiki/Fibonacci_number

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.
Padovan_triangles_(1)
Image Source : https://en.wikipedia.org/wiki/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;
}

Python

# 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)


Output:

21

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