Hexanacci Numbers

The hexanacci number is a generalization of the Fibonacci number where each term is the sum of the six preceding terms. The first few Hexanacci numbers are as follows –  0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840…..

Nth Term of Hexanacci number is given by :

T(n) = T(n-1) + T(n-2) + T(n-3) + T(n-4) + T(n-5) + T(n-6)

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

Find the N^th term of the Hexanacci number

Given a number N. The task is to find the Nth Hexanacci number.

Examples:

Input: N = 8 
Output: 2

Input: N = 11 
Output: 16 

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

Below is the implementation of the above approach: 

C++

// C++ simple recursive program to print
// Nth Hexanacci numbers.
#include<bits/stdc++.h>
#include<iostream>
using namespace std;

// Function to print the Nth
// Hexanacci number
int printhexaRec(int n)
{
    if (n == 0 || n == 1 ||
        n == 2 || n == 3 ||
        n == 4 || n == 5)
        return 0;

    else if (n == 6)
        return 1;

    else
        return (printhexaRec(n - 1) +
                printhexaRec(n - 2) +
                printhexaRec(n - 3) +
                printhexaRec(n - 4) +
                printhexaRec(n - 5) +
                printhexaRec(n - 6));
}

int printhexa(int n)
{
    cout << printhexaRec(n) << endl;
}

// Driver code
int main()
{
    privatenthexa(n);
}

// This code is contributed by Ritik Bansal

Java

// Java simple recursive program to print
// Nth Hexanacci numbers.
import java.util.*;
class GFG{

// Function to print the Nth
// Hexanacci number
static int printhexaRec(int n)
{
    if (n == 0 || n == 1 ||
        n == 2 || n == 3 ||
        n == 4 || n == 5)
        return 0;

    else if (n == 6)
        return 1;

    else
        return (printhexaRec(n - 1) +
                printhexaRec(n - 2) +
                printhexaRec(n - 3) +
                printhexaRec(n - 4) +
                printhexaRec(n - 5) +
                printhexaRec(n - 6));
}

static void printhexa(int n)
{
    System.out.print(printhexaRec(n) + "\n");
}

// Driver code
public static void main(String[] args)
{
    int n = 11;
    printhexa(n);
}
}

// This code is contributed by 29AjayKumar

Python3

# Python simple recursive program to print
# Nth Hexanacci numbers.

# Function to print the Nth
# Hexanacci number
def printhexaRec(n) :
    if (n == 0 or n == 1 or\
        n == 2 or n == 3 or\
        n == 4 or n == 5):
        return 0
    elif (n == 6):
        return 1
    else :
        return (printhexaRec(n - 1) +
                printhexaRec(n - 2) +
                printhexaRec(n - 3)+
                printhexaRec(n - 4)+
                printhexaRec(n - 5)+
                printhexaRec(n - 6))

def printhexa(n) :
    print(printhexaRec(n))


# Driver code
n = 11
printhexa(n)

C#

// C# simple recursive program to print
// Nth Hexanacci numbers.
using System;
class GFG{

// Function to print the Nth
// Hexanacci number
static int printhexaRec(int n)
{
    if (n == 0 || n == 1 ||
        n == 2 || n == 3 ||
        n == 4 || n == 5)
        return 0;

    else if (n == 6)
        return 1;

    else
        return (printhexaRec(n - 1) +
                printhexaRec(n - 2) +
                printhexaRec(n - 3) +
                printhexaRec(n - 4) +
                printhexaRec(n - 5) +
                printhexaRec(n - 6));
}

static void printhexa(int n)
{
    Console.Write(printhexaRec(n) + "\n");
}

// Driver code
public static void Main()
{
    int n = 11;
    printhexa(n);
}
}

// This code is contributed by Code_Mech

Javascript

<script>

    // Javascript simple recursive program to print
    // Nth Hexanacci numbers.

    // Function to print the Nth
    // Hexanacci number
    function printhexaRec(n)
    {
        if (n == 0 || n == 1 ||
            n == 2 || n == 3 ||
            n == 4 || n == 5)
            return 0;

        else if (n == 6)
            return 1;

        else
            return (printhexaRec(n - 1) +
                    printhexaRec(n - 2) +
                    printhexaRec(n - 3) +
                    printhexaRec(n - 4) +
                    printhexaRec(n - 5) +
                    printhexaRec(n - 6));
    }

    function printhexa(n)
    {
        document.write(printhexaRec(n) + "</br>");
    }

    let n = 11;
    printhexa(n);

</script>

Output:

16

Efficient Approach: A better solution is to use Dynamic Programming The idea is to keep track of the last six terms of the to compute the last current term of the series. Initialize the last six terms of the series as:

T(0) = T(1) = T(2) = T(3) = T(4) = 0, T(5) = 1

Finally, compute the terms up to the N^th term of the series:

Below is the implementation of the above approach:

C++

// C++ implementation to print
// Nth term of hexanacci numbers.
#include <bits/stdc++.h>
using namespace std;

// Function to print the Nth
// term of the Hexanacci number
void printhexa(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 = 0;
    int sixth = 1;

    // Declare a current variable
    int curr = 0;

    if (n < 6)
        cout << first << endl;
    else if (n == 6)
        cout << sixth << endl;
    else
    {

        // Loop to add previous five numbers
        // for each number starting from 5
        // and then assign first, second,
        // third, fourth fifth to second,
        // third, fourth, fifth
        // and curr to sixth respectively
        for(int i = 6; i < n; i++)
        {
            curr = first + second + third +
                   fourth + fifth + sixth;

            first = second;
            second = third;
            third = fourth;
            fourth = fifth;
            fifth = sixth;
            sixth = curr;
        }
    }
    cout << curr << endl;
}

// Driver Code
int main()
{
    int n = 11;

    printhexa(n);

    return 0;
}

// This code is contributed by divyeshrabadiya07

Java

// Java implementation to print
// Nth term of hexanacci numbers.
class GFG {

// Function to print the Nth
// term of the Hexanacci number
static void printhexa(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 = 0;
    int sixth = 1;

    // Declare a current variable
    int curr = 0;

    if (n < 6)
        System.out.println(first);
    else if (n == 6)
        System.out.println(sixth);

    else
    {

        // Loop to add previous five numbers
        // for each number starting from 5
        // and then assign first, second,
        // third, fourth fifth to second,
        // third, fourth, fifth
        // and curr to sixth respectively
        for(int i = 6; i < n; i++)
        {
            curr = first + second + third +
                   fourth + fifth + sixth;
            first = second;
            second = third;
            third = fourth;
            fourth = fifth;
            fifth = sixth;
            sixth = curr;
        }
    }
    System.out.println(curr);
}

// Driver code
public static void main(String[] args)
{
    int n = 11;

    printhexa(n);
}
}

// This code is contributed by Amit Katiyar

Python3

# Python3 implementation to print
# Nth term of hexanacci numbers.

# Function to print the Nth
# term of the Hexanacci number
def printhexa(n) :
    if (n < 0):
        return

    # Initialize first five
    # numbers to base cases
    first = 0
    second = 0
    third = 0
    fourth = 0
    fifth = 0
    sixth = 1

    # declare a current variable
    curr = 0

    if (n <6):
        print(first)
    elif (n == 6):
        print(sixth)

    else:

        # Loop to add previous five numbers
        # for each number starting from 5
        # and then assign first, second,
        # third, fourth fifth to second,
        # third, fourth, fifth
        # and curr to sixth respectively
        for i in range(6, n):
            curr = first + second + third +\
                   fourth + fifth + sixth
            first = second
            second = third
            third = fourth
            fourth = fifth
            fifth = sixth
            sixth = curr

    print(curr)

# Driver code
n = 11
printhexa(n)

C#

// C# implementation to print
// Nth term of hexanacci numbers.
using System;

class GFG{

// Function to print the Nth
// term of the Hexanacci number
static void printhexa(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 = 0;
    int sixth = 1;

    // Declare a current variable
    int curr = 0;

    if (n < 6)
        Console.WriteLine(first);
    else if (n == 6)
        Console.WriteLine(sixth);

    else
    {

        // Loop to add previous five numbers
        // for each number starting from 5
        // and then assign first, second,
        // third, fourth fifth to second,
        // third, fourth, fifth
        // and curr to sixth respectively
        for(int i = 6; i < n; i++)
        {
            curr = first + second + third +
                   fourth + fifth + sixth;
            first = second;
            second = third;
            third = fourth;
            fourth = fifth;
            fifth = sixth;
            sixth = curr;
        }
    }
    Console.WriteLine(curr);
}

// Driver code
public static void Main(String[] args)
{
    int n = 11;

    printhexa(n);
}
}

// This code is contributed by Amit Katiyar

Javascript

<script>
    // Javascript implementation to print
    // Nth term of hexanacci numbers.

    // Function to print the Nth
    // term of the Hexanacci number
    function printhexa(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 = 0;
        let sixth = 1;

        // Declare a current variable
        let curr = 0;

        if (n < 6)
            document.write(first);
        else if (n == 6)
            document.write(sixth);
        else
        {

            // Loop to add previous five numbers
            // for each number starting from 5
            // and then assign first, second,
            // third, fourth fifth to second,
            // third, fourth, fifth
            // and curr to sixth respectively
            for(let i = 6; i < n; i++)
            {
                curr = first + second + third +
                       fourth + fifth + sixth;

                first = second;
                second = third;
                third = fourth;
                fourth = fifth;
                fifth = sixth;
                sixth = curr;
            }
        }
        document.write(curr);
    }
    let n = 11;
    printhexa(n);

    // This code is contributed by divyesh072019.
</script>

Output:

16

Time Complexity: O(N)

Auxiliary Space: O(1)