Pentanacci Numbers

Last Updated : 27 Feb, 2022

The Pentanacci series is a generalization of the Fibonacci sequence where each term is the sum of the five preceding terms. The first few Pentanacci numbers are as follows – 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519…..

Nth Term of Pentanacci number is given by:

T(n) = T(n-1) + T(n-2) + T(n-3) + T(n-4) + T(n-5)
with T(0) = T(1) = T(2) = T(3) = 0, T(4) = 1

Find the N^th Pentanacci number

Given a number N. The task is to find the N-th Pentanacci number.

Examples:

Input: N = 7
Output: 2

Input: N = 10
Output: 16

Naive Approach: The idea is to follow the recurrence for finding the number and use recursion to solve it.

Recurrence relation:
T(n) = T(n-1) + T(n-2) + T(n-3) + T(n-4) + T(n-5)

Below is the implementation of the above approach:

C++14

   // A simple recursive program to print 
// Nth Pentanacci number 
#include<bits/stdc++.h>
using namespace std;
     
// Recursive function to find the Nth
// Pentanacci number
int printpentaRec(int n)
{
    if (n == 0 || n == 1 ||
        n == 2 || n == 3)
        return 0;
         
    else if (n == 4 || n == 5)
        return 1;
    else
        return (printpentaRec(n - 1) +
                printpentaRec(n - 2) +
                printpentaRec(n - 3)+
                printpentaRec(n - 4)+
                printpentaRec(n - 5));
}
             
// Function to print the Nth
// Pentanacci number
void printPenta(int n)
{
    cout << printpentaRec(n) << "\n";
}
             
// Driver code 
int main()
{
    int n = 10;
     
    printPenta(n);
     
    return 0;
}
 
// This code is contributed by yatinagg
 
  

Java

   // A simple recursive program to print 
// Nth Pentanacci number 
import java.util.*;
class GFG{
     
// Recursive function to find the Nth
// Pentanacci number
static int printpentaRec(int n)
{
    if (n == 0 || n == 1 ||
        n == 2 || n == 3)
        return 0;
         
    else if (n == 4 || n == 5)
        return 1;
    else
        return (printpentaRec(n - 1) +
                printpentaRec(n - 2) +
                printpentaRec(n - 3) +
                printpentaRec(n - 4) +
                printpentaRec(n - 5));
}
             
// Function to print the Nth
// Pentanacci number
static void printPenta(int n)
{
    System.out.print(printpentaRec(n) + "\n");
}
             
// Driver code 
public static void main(String[] args)
{
    int n = 10;
     
    printPenta(n); 
}
}
 
// This code is contributed by gauravrajput1
 
  

Python3

   # A simple recursive program to print 
# Nth Pentanacci number 
   
# Recursive program to find the Nth
# Pentanacci number
def printpentaRec(n) : 
    if (n == 0 or n == 1 or\
        n == 2 or n == 3): 
        return 0
    elif (n == 4 or n == 5): 
        return 1
    else : 
        return (printpentaRec(n - 1) +
                printpentaRec(n - 2) +
                printpentaRec(n - 3)+
                printpentaRec(n - 4)+
                printpentaRec(n - 5)) 
           
# Function to print the Nth
# Pentanacci number
def printPenta(n) : 
    print(printpentaRec(n)) 
           
   
# Driver code 
n = 10
printPenta(n)
 
  

C#

   // A simple recursive program to print 
// Nth Pentanacci number 
using System;
 
class GFG{
     
// Recursive function to find the Nth
// Pentanacci number
static int printpentaRec(int n)
{
    if (n == 0 || n == 1 ||
        n == 2 || n == 3)
        return 0;
          
    else if (n == 4 || n == 5)
        return 1;
    else
        return (printpentaRec(n - 1) +
                printpentaRec(n - 2) +
                printpentaRec(n - 3) +
                printpentaRec(n - 4) +
                printpentaRec(n - 5));
}
              
// Function to print the Nth
// Pentanacci number
static void printPenta(int n)
{
    Console.WriteLine(printpentaRec(n));
}
 
// Driver code
static void Main() 
{
    int n = 10;
      
    printPenta(n); 
}
}
 
// This code is contributed divyeshrabadiya07
 
  

Javascript

   <script>
    // A simple recursive program to print 
    // Nth Pentanacci number 
     
    // Recursive function to find the Nth
    // Pentanacci number
    function printpentaRec(n)
    {
        if (n == 0 || n == 1 ||
            n == 2 || n == 3)
            return 0;
        else if (n == 4 || n == 5)
            return 1;
        else
            return (printpentaRec(n - 1) +
                    printpentaRec(n - 2) +
                    printpentaRec(n - 3)+
                    printpentaRec(n - 4)+
                    printpentaRec(n - 5));
    }
 
    // Function to print the Nth
    // Pentanacci number
    function printPenta(n)
    {
        document.write(printpentaRec(n) + "</br>");
    }
 
    let n = 10;     
    printPenta(n);
 
// This code is contributed by divyesh072019.
</script>
 
  

Output:

Efficient Approach: The idea is to use Dynamic Programming to solve this problem. That is memoization of the solution in four variables for the last four terms such that the same subproblem is not computed again and again.

Below is the implementation of the above approach:

C++14

   // C++14 implementation to print 
// Nth Pentanacci numbers. 
#include<bits/stdc++.h>
using namespace std;
     
// Function to print Nth 
// Pentanacci number 
void printpenta(int n)
{
    if (n < 0) 
        return;
     
    // Initialize first five 
    // numbers to base cases 
    int first = 0;
    int second = 0;
    int third = 0;
    int fourth = 0;
    int fifth = 1;
     
    // Declare a current variable 
    int curr = 0;
     
    if (n == 0 || n == 1 ||
        n == 2 || n == 3)
        cout << first << "\n";
         
    else if (n == 5)
        cout << fifth << "\n"; 
     
    else
    {
     
        // Loop to add previous five numbers 
        // for each number starting from 5 
        // and then assign first, second, 
        // third, fourth to second, third, fourth 
        // and curr to fifth respectively 
        for(int i = 5; i < n; i++)
        {
            curr = first + second +
                   third + fourth + fifth;
            first = second;
            second = third; 
            third = fourth; 
            fourth = fifth;
            fifth = curr;
        }
    cout << curr << "\n";
    }
}
             
// Driver code 
int main()
{
    int n = 10;
     
    printpenta(n);
     
    return 0;
}
 
// This code is contributed by yatinagg
 
  

Java

   // Java implementation to print 
// Nth Pentanacci numbers. 
import java.util.*;
class GFG{
     
// Function to print Nth 
// Pentanacci number 
static void printpenta(int n)
{
  if (n < 0) 
    return;
 
  // Initialize first five 
  // numbers to base cases 
  int first = 0;
  int second = 0;
  int third = 0;
  int fourth = 0;
  int fifth = 1;
 
  // Declare a current variable 
  int curr = 0;
 
  if (n == 0 || n == 1 ||
      n == 2 || n == 3)
    System.out.print(first + "\n");
  else if (n == 5)
    System.out.print(fifth + "\n"); 
  else
  {
    // Loop to add previous five numbers 
    // for each number starting from 5 
    // and then assign first, second, 
    // third, fourth to second, third, 
    // fourth and curr to fifth respectively 
    for(int i = 5; i < n; i++)
    {
      curr = first + second +
             third + fourth + fifth;
      first = second;
      second = third; 
      third = fourth; 
      fourth = fifth;
      fifth = curr;
    }
    System.out.print(curr + "\n");
  }
}
             
// Driver code 
public static void main(String[] args)
{
  int n = 10;
  printpenta(n);    
}
}
 
// This code is contributed by Princi Singh
 
  

Python3

   # Python3 implementation to print 
# Nth Pentanacci numbers. 
   
# Function to print Nth 
# Pentanacci number      
def printpenta(n) : 
    if (n < 0):  
        return 
   
    # Initialize first five  
    # numbers to base cases  
    first = 0 
    second = 0 
    third = 0 
    fourth = 0
    fifth = 1
   
    # declare a current variable  
    curr = 0 
   
    if (n == 0  or n == 1 or\
        n == 2 or n == 3):  
        print(first)
    elif (n == 5):  
        print(fifth)  
   
    else: 
   
        # Loop to add previous five numbers  
        # for each number starting from 5  
        # and then assign first, second,  
        # third, fourth to second, third, fourth  
        # and curr to fifth respectively  
        for i in range(5, n): 
            curr = first + second +\
                 third + fourth + fifth
            first = second  
            second = third  
            third = fourth  
            fourth = fifth
            fifth = curr  
           
    print(curr)   
   
# Driver code 
n = 10
printpenta(n)
 
  

C#

   // C# implementation to print 
// Nth Pentanacci numbers. 
using System;
class GFG{
     
// Function to print Nth 
// Pentanacci number 
static void printpenta(int n)
{
  if (n < 0) 
    return;
 
  // Initialize first five 
  // numbers to base cases 
  int first = 0;
  int second = 0;
  int third = 0;
  int fourth = 0;
  int fifth = 1;
 
  // Declare a current variable 
  int curr = 0;
 
  if (n == 0 || n == 1 ||
      n == 2 || n == 3)
    Console.Write(first + "\n");
  else if (n == 5)
    Console.Write(fifth + "\n"); 
  else
  {
    // Loop to add previous five numbers 
    // for each number starting from 5 
    // and then assign first, second, 
    // third, fourth to second, third, 
    // fourth and curr to fifth respectively 
    for(int i = 5; i < n; i++)
    {
      curr = first + second +
             third + fourth + fifth;
      first = second;
      second = third; 
      third = fourth; 
      fourth = fifth;
      fifth = curr;
    }
    Console.Write(curr + "\n");
  }
}
 
// Driver code 
public static void Main(String[] args)
{
  int n = 10;
  printpenta(n);    
}
}
 
// This code is contributed by shikhasingrajput
 
  

Javascript

   <script>
 
    // Javascript implementation to print
    // Nth Pentanacci numbers.
     
    // Function to print Nth
    // Pentanacci number
    function printpenta(n)
    {
      if (n < 0)
        return;
 
      // Initialize first five
      // numbers to base cases
      let first = 0;
      let second = 0;
      let third = 0;
      let fourth = 0;
      let fifth = 1;
 
      // Declare a current variable
      let curr = 0;
 
      if (n == 0 || n == 1 ||
          n == 2 || n == 3)
        document.write(first + "</br>");
      else if (n == 5)
        document.write(fifth + "</br>");
      else
      {
        // Loop to add previous five numbers
        // for each number starting from 5
        // and then assign first, second,
        // third, fourth to second, third,
        // fourth and curr to fifth respectively
        for(let i = 5; i < n; i++)
        {
          curr = first + second +
                 third + fourth + fifth;
          first = second;
          second = third;
          third = fourth;
          fourth = fifth;
          fifth = curr;
        }
        document.write(curr + "</br>");
      }
    }
     
    let n = 10;
      printpenta(n);  
     
</script>
 
  

Output:

Pentatope Numbers

SHUBHAMSINGH10

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Pentanacci Numbers

Find the Nth Pentanacci number

C++14

Java

Python3

C#

Javascript

C++14

Java

Python3

C#

Javascript

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Find the N^th Pentanacci number