Tetranacci Numbers

The tetranacci numbers are a generalization of the Fibonacci numbers defined by the recurrence relation 

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

For n>=4. They represent the n=4 case of the Fibonacci n-step numbers. The first few terms for n=0, 1, … are 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, … 
Given a number N. The task is to find the N-th tetranacci number.

Examples: 

Input: 5
Output: 4

Input: 9
Output: 108 

A naive approach is to follow the recurrence for finding the number and use recursion to solve it.
Below is the implementation of the above approach. 

C++

// A simple recursive CPP program to print
// the nth tetranacci numbers.
#include <iostream>
using namespace std;

// Function to return the
// N-th tetranacci number
int printTetraRec(int n)
{
    // base cases
    if (n == 0)
        return 0;
    // base cases
    if (n == 1 || n == 2)
        return 1;
    // base cases
    if (n == 3)
        return 2;

    else
        return printTetraRec(n - 1) + printTetraRec(n - 2)
               + printTetraRec(n - 3) + printTetraRec(n - 4);
}

// function to print the nth tetranacci number
void printTetra(int n)
{
    cout << printTetraRec(n) << " ";
}

// Driver code
int main()
{
    int n = 10;
    printTetra(n);
    return 0;
}

Java

// A simple recursive Java
// program to print the nth
// tetranacci numbers.
class GFG
{
// Function to return the
// N-th tetranacci number
static int printTetraRec(int n)
{
    // base cases
    if (n == 0)
        return 0;
    // base cases
    if (n == 1 || n == 2)
        return 1;
    // base cases
    if (n == 3)
        return 2;

    else
        return printTetraRec(n - 1) +
               printTetraRec(n - 2) +
               printTetraRec(n - 3) +
               printTetraRec(n - 4);
}

// function to print the
// Nth tetranacci number
static void printTetra(int n)
{
    System.out.println(printTetraRec(n) + " ");
}

// Driver code
public static void main(String[] args)
{
    int n = 10;
    printTetra(n);
}
}

// This code is contributed by mits

Python3

# A simple recursive Python3 program
# to print the nth tetranacci numbers.

# Function to return the
# N-th tetranacci number
def printTetraRec(n):

    # base cases
    if (n == 0):
        return 0;

    # base cases
    if (n == 1 or n == 2):
        return 1;

    # base cases
    if (n == 3):
        return 2;

    else:
        return (printTetraRec(n - 1) +
                printTetraRec(n - 2) +
                printTetraRec(n - 3) +
                printTetraRec(n - 4));

# function to print the
# nth tetranacci number
def printTetra(n):
    print(printTetraRec(n), end = " ");

# Driver code
n = 10;
printTetra(n);

# This code is contributed
# by mits

C#

// A simple recursive C#
// program to print the nth
// tetranacci numbers.
class GFG
{

// Function to return the
// N-th tetranacci number
static int printTetraRec(int n)
{
    // base cases
    if (n == 0)
        return 0;
    // base cases
    if (n == 1 || n == 2)
        return 1;
    // base cases
    if (n == 3)
        return 2;

    else
        return printTetraRec(n - 1) +
               printTetraRec(n - 2) +
               printTetraRec(n - 3) +
               printTetraRec(n - 4);
}

// function to print the
// Nth tetranacci number
static void printTetra(int n)
{
    System.Console.WriteLine(
           printTetraRec(n) + " ");
}

// Driver code
static void Main()
{
    int n = 10;
    printTetra(n);
}
}

// This code is contributed by mits

PHP

<?php
// A simple recursive PHP program
// to print the nth tetranacci numbers.

// Function to return the
// N-th tetranacci number
function printTetraRec($n)
{
    // base cases
    if ($n == 0)
        return 0;

    // base cases
    if ($n == 1 || $n == 2)
        return 1;

    // base cases
    if ($n == 3)
        return 2;

    else
        return printTetraRec($n - 1) +
               printTetraRec($n - 2) +
               printTetraRec($n - 3) +
               printTetraRec($n - 4);
}

// function to print the
// nth tetranacci number
function printTetra($n)
{
    echo printTetraRec($n) . " ";
}

// Driver code
$n = 10;
printTetra($n);

// This code is contributed
// by Abby_akku
?>

Javascript

<script>

    // A simple recursive Javascript
    // program to print the nth
    // tetranacci numbers.

    // Function to return the
    // N-th tetranacci number
    function printTetraRec(n)
    {
        // base cases
        if (n == 0)
            return 0;
        // base cases
        if (n == 1 || n == 2)
            return 1;
        // base cases
        if (n == 3)
            return 2;

        else
            return printTetraRec(n - 1) +
                   printTetraRec(n - 2) +
                   printTetraRec(n - 3) +
                   printTetraRec(n - 4);
    }

    // function to print the
    // Nth tetranacci number
    function printTetra(n)
    {
        document.write(printTetraRec(n) + " " + "</br>");
    }

    let n = 10;
    printTetra(n);

</script>

Output:

208

Time Complexity: O(4^N) 
Auxiliary Space: O(4^N), The extra space is used due to the recursion call stack.

A better solution is to use Dynamic Programming (memoization) as there are multiple overlaps.
Given below is the recursive tree for N=10. 

                                        rec(10)

                             /         /       \          \

                       rec(9)      rec(8)     rec(7)     rec(6)

              /      /     \     \
                     
          rec(8) rec(7)  rec(6)  rec(5)

In the above partial recursion tree, rec(8), rec(7), rec(6) have been solved twice. In drawing the complete recursion tree, it has been observed that there are many subproblems that are solved again and again. So this problem has overlapping Substructure property and recomputation of same subproblems can be avoided by either using Memoization or Tabulation.
Below is the implementation of the above approach.  

C++

// A DP based CPP
// program to print
// the nth tetranacci number
#include <iostream>
using namespace std;

// Function to print the
// N-th tetranacci number
int printTetra(int n)
{
    int dp[n + 5];
    // base cases
    dp[0] = 0;
    dp[1] = dp[2] = 1;
    dp[3] = 2;

    for (int i = 4; i <= n; i++)
        dp[i] = dp[i - 1] + dp[i - 2] +
                dp[i - 3] + dp[i - 4];

    cout << dp[n];
}

// Driver code
int main()
{
    int n = 10;
    printTetra(n);
    return 0;
}

Java

// A DP based Java
// program to print
// the nth tetranacci number

class GFG{
// Function to print the
// N-th tetranacci number
static void printTetra(int n)
{
    int[] dp=new int[n + 5];
    // base cases
    dp[0] = 0;
    dp[1] = dp[2] = 1;
    dp[3] = 2;

    for (int i = 4; i <= n; i++)
        dp[i] = dp[i - 1] + dp[i - 2] +
                dp[i - 3] + dp[i - 4];

    System.out.print(dp[n]);
}

// Driver code
public static void main(String[] args)
{
    int n = 10;
    printTetra(n);
}
}
// This code is contributed by mits

Python3

# A DP based Python3 program to print
# the nth tetranacci number

# Function to print the
# N-th tetranacci number
def printTetra(n):
    dp = [0] * (n + 5);

    # base cases
    dp[0] = 0;
    dp[1] = 1;
    dp[2] = 1;
    dp[3] = 2;

    for i in range(4, n + 1):
        dp[i] = (dp[i - 1] + dp[i - 2] +
                 dp[i - 3] + dp[i - 4]);

    print(dp[n]);

# Driver code
n = 10;
printTetra(n);

# This code is contributed by mits

C#

// A DP based C#
// program to print
// the nth tetranacci number

class GFG{
// Function to print the
// N-th tetranacci number
static void printTetra(int n)
{
    int[] dp=new int[n + 5];
    // base cases
    dp[0] = 0;
    dp[1] = dp[2] = 1;
    dp[3] = 2;

    for (int i = 4; i <= n; i++)
        dp[i] = dp[i - 1] + dp[i - 2] +
                dp[i - 3] + dp[i - 4];

    System.Console.WriteLine(dp[n]);
}

// Driver code
static void Main()
{
    int n = 10;
    printTetra(n);
}
}
// This code is contributed by mits

PHP

<?php
// A DP based PHP
// program to print
// the nth tetranacci number

// Function to print the
// N-th tetranacci number
function printTetra($n)
{
    $dp = array_fill(0, $n + 5, 0);

    // base cases
    $dp[0] = 0;
    $dp[1] = $dp[2] = 1;
    $dp[3] = 2;

    for ($i = 4; $i <= $n; $i++)
        $dp[$i] = $dp[$i - 1] + $dp[$i - 2] +
                  $dp[$i - 3] + $dp[$i - 4];

    echo $dp[$n];
}

// Driver code
$n = 10;
printTetra($n);

// This code is contributed by mits
?>

Javascript

<script>
    // A DP based Javascript
    // program to print
    // the nth tetranacci number

    // Function to print the
    // N-th tetranacci number
    function printTetra(n)
    {
        let dp=new Array(n + 5);
        // base cases
        dp[0] = 0;
        dp[1] = dp[2] = 1;
        dp[3] = 2;

        for (let i = 4; i <= n; i++)
            dp[i] = dp[i - 1] + dp[i - 2] +
                    dp[i - 3] + dp[i - 4];

        document.write(dp[n]);
    }

    let n = 10;
    printTetra(n);

</script>

Output:

208

Time Complexity: O(N) 
Auxiliary Space: O(N)

The time complexity above is linear, but it requires extra space. Space used can be optimized in the above solution by using four variables to keep track of the previous four numbers.
Below is the implementation of the above approach. 

C++

// A space optimized
// based CPP program to
// print the nth tetranacci number
#include <iostream>
using namespace std;

// Function to print the
// N-th tetranacci number
void printTetra(int n)
{
    if (n < 0)
        return;

    // Initialize first
    // four numbers to base cases
    int first = 0, second = 1;
    int third = 1, fourth = 2;

    // declare a current variable
    int curr;

    if (n == 0)
        cout << first;
    else if (n == 1 || n == 2)
        cout << second;

    else if (n == 3)
        cout << fourth;

    else {

        // Loop to add previous
        // four numbers for
        // each number starting
        // from 4 and then assign
        // first, second, third
        // to second, third, fourth and
        // curr to fourth respectively
        for (int i = 4; i <= n; i++) {
            curr = first + second + third + fourth;
            first = second;
            second = third;
            third = fourth;
            fourth = curr;
        }
        cout << curr;
    }
}

// Driver code
int main()
{
    int n = 10;
    printTetra(n);
    return 0;
}

Java

// A space optimized
// based Java program to
// print the nth tetranacci number
import java.io.*;
import java.util.*;
import java.lang.*;

class GFG{
// Function to print the
// N-th tetranacci number
static void printTetra(int n)
{
    if (n < 0)
        return;

    // Initialize first
    // four numbers to base cases
    int first = 0, second = 1;
    int third = 1, fourth = 2;

    // declare a current variable
    int curr = 0;

    if (n == 0)
        System.out.print(first);
    else if (n == 1 || n == 2)
        System.out.print(second);

    else if (n == 3)
        System.out.print(fourth);

    else
    {

        // Loop to add previous
        // four numbers for
        // each number starting
        // from 4 and then assign
        // first, second, third
        // to second, third, fourth and
        // curr to fourth respectively
        for (int i = 4; i <= n; i++)
        {
            curr = first + second + third + fourth;
            first = second;
            second = third;
            third = fourth;
            fourth = curr;
        }
        System.out.print(curr);
    }
}

// Driver code
public static void main(String[] args)
{
    int n = 10;
    printTetra(n);
}
}

// This code is contributed
// by Akanksha Rai(Abby_akku)

Python3

# A space optimized based Python3 program
# to print the nth tetranacci number

# Function to print the N-th
# tetranacci number
def printTetra(n):

    if (n < 0):
        return;

    # Initialize first four
    # numbers to base cases
    first = 0;
    second = 1;
    third = 1;
    fourth = 2;

    # declare a current variable
    curr = 0;

    if (n == 0):
        print(first);
    elif (n == 1 or n == 2):
        print(second);

    elif (n == 3):
        print(fourth);

    else:

        # Loop to add previous four numbers
        # for each number starting from 4
        # and then assign first, second,
        # third to second, third, fourth
        # and curr to fourth respectively
        for i in range(4, n + 1):
            curr = first + second + third + fourth;
            first = second;
            second = third;
            third = fourth;
            fourth = curr;

    print(curr);

# Driver code
n = 10;
printTetra(n);

# This code is contributed by mits

C#

// A space optimized based C# program to
// print the nth tetranacci number
using System;

class GFG{

// Function to print the
// N-th tetranacci number
static void printTetra(int n)
{
    if (n < 0)
        return;

    // Initialize first
    // four numbers to base cases
    int first = 0, second = 1;
    int third = 1, fourth = 2;

    // declare a current variable
    int curr = 0;

    if (n == 0)
        Console.Write(first);
    else if (n == 1 || n == 2)
        Console.Write(second);

    else if (n == 3)
        Console.Write(fourth);

    else
    {

        // Loop to add previous
        // four numbers for
        // each number starting
        // from 4 and then assign
        // first, second, third
        // to second, third, fourth and
        // curr to fourth respectively
        for (int i = 4; i <= n; i++)
        {
            curr = first + second + third + fourth;
            first = second;
            second = third;
            third = fourth;
            fourth = curr;
        }
        Console.Write(curr);
    }
}

    // Driver code
    static public void Main ()
    {

        int n = 10;
        printTetra(n);
    }
}

// This code is contributed ajit

PHP

<?php
// A space optimized based PHP program
// to print the nth tetranacci number

// Function to print the N-th
// tetranacci number

function printTetra($n)
{
    if ($n < 0)
        return;

    // Initialize first four
    // numbers to base cases
    $first = 0;
    $second = 1;
    $third = 1;
    $fourth = 2;

    // declare a current variable
    $curr;

    if ($n == 0)
        echo $first;
    else if ($n == 1 || $n == 2)
        echo $second;

    else if ($n == 3)
        echo $fourth;

    else
    {

        // Loop to add previous four
        // numbers for each number
        // starting from 4 and then
        // assign first, second, third
        // to second, third, fourth and
        // curr to fourth respectively
        for ($i = 4; $i <= $n; $i++)
        {
            $curr = $first + $second +
                    $third + $fourth;
            $first = $second;
            $second = $third;
            $third = $fourth;
            $fourth = $curr;
        }
    echo $curr;
    }
}

// Driver code
$n = 10;
printTetra($n);

// This code is contributed by ajit
?>

Javascript

<script>

// A space optimized
// based Javascript program to
// print the nth tetranacci number

// Function to print the
// N-th tetranacci number
function printTetra(n)
{
    if (n < 0)
        return;

    // Initialize first
    // four numbers to base cases
    var first = 0, second = 1;
    var third = 1, fourth = 2;

    // declare a current variable
    var curr;

    if (n == 0)
        cout << first;
    else if (n == 1 || n == 2)
        cout << second;

    else if (n == 3)
        cout << fourth;

    else {

        // Loop to add previous
        // four numbers for
        // each number starting
        // from 4 and then assign
        // first, second, third
        // to second, third, fourth and
        // curr to fourth respectively
        for (var i = 4; i <= n; i++) {
            curr = first + second + third + fourth;
            first = second;
            second = third;
            third = fourth;
            fourth = curr;
        }
        document.write( curr);
    }
}

// Driver code
var n = 10;
printTetra(n);

</script>

Output:

208

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