Count ways to reach the nth stair using step 1, 2 or 3

A child is running up a staircase with n steps and can hop either 1 step, 2 steps, or 3 steps at a time. Implement a method to count how many possible ways the child can run up the stairs.

Examples:

Input : 4
Output : 7
Explanation:
Below are the seven ways
 1 step + 1 step + 1 step + 1 step
 1 step + 2 step + 1 step
 2 step + 1 step + 1 step 
 1 step + 1 step + 2 step
 2 step + 2 step
 3 step + 1 step
 1 step + 3 step

Input : 3
Output : 4
Explanation:
Below are the four ways
 1 step + 1 step + 1 step
 1 step + 2 step
 2 step + 1 step
 3 step

There are two methods to solve this problem:

1. Recursive Method
2. Dynamic Programming

Method 1: Recursive. 
There are n stairs, and a person is allowed to jump next stair, skip one stair or skip two stairs. So there are n stairs. So if a person is standing at i-th stair, the person can move to i+1, i+2, i+3-th stair. A recursive function can be formed where at current index i the function is recursively called for i+1, i+2 and i+3 th stair. 
There is another way of forming the recursive function. To reach a stair i, a person has to jump either from i-1, i-2 or i-3 th stair or i is the starting stair.

Algorithm:

1. Create a recursive function (count(int n)) which takes only one parameter.
2. Check the base cases. If the value of n is less than 0 then return 0, and if the value of n is equal to zero then return 1 as it is the starting stair.
3. Call the function recursively with values n-1, n-2 and n-3 and sum up the values that are returned, i.e. sum = count(n-1) + count(n-2) + count(n-3)
4. Return the value of the sum.

C++

// C++ Program to find n-th stair using step size
// 1 or 2 or 3.
#include <iostream>
using namespace std;

class GFG {

    // Returns count of ways to reach n-th stair
    // using 1 or 2 or 3 steps.
public:
    int findStep(int n)
    {
        if (n == 0)
            return 1;
        else if (n < 0)
            return 0;

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

// Driver code
int main()
{
    GFG g;
    int n = 4;
    cout << g.findStep(n);
    return 0;
}

// This code is contributed by SoM15242

C

// Program to find n-th stair using step size
// 1 or 2 or 3.
#include <stdio.h>

// Returns count of ways to reach n-th stair
// using 1 or 2 or 3 steps.
int findStep(int n)
{
    if (n == 0)
       return 1;
    else if (n < 0)
        return 0;

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

// Driver code
int main()
{
    int n = 4;
    printf("%d\n", findStep(n));
    return 0;
}

Java

// Program to find n-th stair
// using step size 1 or 2 or 3.
import java.lang.*;
import java.util.*;

public class GfG {

    // Returns count of ways to reach
    // n-th stair using 1 or 2 or 3 steps.
    public static int findStep(int n)
    {
        if ( n == 0)
            return 1;
        else if (n < 0)
            return 0;

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

    // Driver function
    public static void main(String argc[])
    {
        int n = 4;
        System.out.println(findStep(n));
    }
}

/* This code is contributed by Sagar Shukla */

Python

# Python program to find n-th stair
# using step size 1 or 2 or 3.

# Returns count of ways to reach n-th
# stair using 1 or 2 or 3 steps.


def findStep(n):
    if ( n == 0 ):
        return 1
    elif (n < 0):
        return 0

    else:
        return findStep(n - 3) + findStep(n - 2) + findStep(n - 1)


# Driver code
n = 4
print(findStep(n))

# This code is contributed by Nikita Tiwari.

C#

// Program to find n-th stair
// using step size 1 or 2 or 3.
using System;

public class GfG {

    // Returns count of ways to reach
    // n-th stair using 1 or 2 or 3 steps.
    public static int findStep(int n)
    {
        if ( n == 0)
            return 1;
        else if (n < 0)
            return 0;

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

    // Driver function
    public static void Main()
    {
        int n = 4;
        Console.WriteLine(findStep(n));
    }
}

/* This code is contributed by vt_m */

PHP

<?php
// PHP Program to find n-th stair
// using step size 1 or 2 or 3.

// Returns count of ways to
// reach n-th stair using
// 1 or 2 or 3 steps.
function findStep($n)
{
    if ( $n == 0)
        return 1;
    else if ($n < 0)
        return 0;

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

// Driver code
$n = 4;
echo findStep($n);

// This code is contributed by m_kit
?>

Javascript

<script>

// JavaScript Program to find n-th stair using step size
// 1 or 2 or 3.


    // Returns count of ways to reach n-th stair
    // using 1 or 2 or 3 steps.
    function findStep(n)
    {
        if (n == 0)
            return 1;
        else if (n < 0)
            return 0;

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


// Driver code

    let n = 4;
     document.write(findStep(n));



// This code is contributed by Surbhi Tyagi.

</script>

Output

7

Working:

Complexity Analysis:

• Time Complexity: O(3ⁿ). 
  The time complexity of the above solution is exponential, a close upper bound will be O(3ⁿ). From each state, 3 recursive function are called. So the upperbound for n states is O(3ⁿ).
• Space Complexity:O(1). 
  As no extra space is required.

Note: The Time Complexity of the program can be optimized using Dynamic Programming.

Method 2: Dynamic Programming.
The idea is similar, but it can be observed that there are n states but the recursive function is called 3 ^ n times. That means that some states are called repeatedly. So the idea is to store the value of states. This can be done in two ways.  

• Top-Down Approach: The first way is to keep the recursive structure intact and just store the value in a HashMap and whenever the function is called again return the value store without computing ().
• Bottom-Up Approach: The second way is to take an extra space of size n and start computing values of states from 1, 2 .. to n, i.e. compute values of i, i+1, i+2 and then use them to calculate the value of i+3.

Algorithm:

1. Create an array of size n + 1 and initialize the first 3 variables with 1, 1, 2. The base cases.
2. Run a loop from 3 to n.
3. For each index i, computer value of ith position as dp[i] = dp[i-1] + dp[i-2] + dp[i-3].
4. Print the value of dp[n], as the Count of the number of ways to reach n th step.

C++

// A C++ program to count number of ways
// to reach n't stair when
#include <iostream>
using namespace std;

// A recursive function used by countWays
int countWays(int n)
{
    int res[n + 1];
    res[0] = 1;
    res[1] = 1;
    res[2] = 2;
    for (int i = 3; i <= n; i++)
        res[i] = res[i - 1] + res[i - 2] + res[i - 3];

    return res[n];
}

// Driver program to test above functions
int main()
{
    int n = 4;
    cout << countWays(n);
    return 0;
}
// This code is contributed by shubhamsingh10

C

// A C program to count number of ways
// to reach n't stair when
#include <stdio.h>

// A recursive function used by countWays
int countWays(int n)
{
    int res[n + 1];
    res[0] = 1;
    res[1] = 1;
    res[2] = 2;
    for (int i = 3; i <= n; i++)
        res[i] = res[i - 1] + res[i - 2] + res[i - 3];

    return res[n];
}

// Driver program to test above functions
int main()
{
    int n = 4;
    printf("%d", countWays(n));
    return 0;
}

Java

// Program to find n-th stair
// using step size 1 or 2 or 3.
import java.lang.*;
import java.util.*;

public class GfG {

    // A recursive function used by countWays
    public static int countWays(int n)
    {
        int[] res = new int[n + 1];
        res[0] = 1;
        res[1] = 1;
        res[2] = 2;

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

        return res[n];
    }

    // Driver function
    public static void main(String argc[])
    {
        int n = 4;
        System.out.println(countWays(n));
    }
}

/* This code is contributed by Sagar Shukla */

Python

# Python program to find n-th stair
# using step size 1 or 2 or 3.

# A recursive function used by countWays


def countWays(n):
    res = [0] * (n + 2)
    res[0] = 1
    res[1] = 1
    res[2] = 2

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

    return res[n]


# Driver code
n = 4
print(countWays(n))


# This code is contributed by Nikita Tiwari.

C#

// Program to find n-th stair
// using step size 1 or 2 or 3.
using System;

public class GfG {

    // A recursive function used by countWays
    public static int countWays(int n)
    {
        int[] res = new int[n + 2];
        res[0] = 1;
        res[1] = 1;
        res[2] = 2;

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

        return res[n];
    }

    // Driver function
    public static void Main()
    {
        int n = 4;
        Console.WriteLine(countWays(n));
    }
}

/* This code is contributed by vt_m */

PHP

<?php
// A PHP program to count
// number of ways to reach
// n'th stair when

// A recursive function
// used by countWays
function countWays($n)
{
    $res[0] = 1;
    $res[1] = 1;
    $res[2] = 2;
    for ($i = 3; $i <= $n; $i++)
        $res[$i] = $res[$i - 1] +
                   $res[$i - 2] +
                   $res[$i - 3];

    return $res[$n];
}

// Driver Code
$n = 4;
echo countWays($n);

// This code is contributed by ajit
?>

Javascript

<script>
    // JavaScript Program to find n-th stair
    // using step size 1 or 2 or 3.

    // A recursive function used by countWays
    function countWays(n)
    {
        let res = new Array(n + 2);
        res[0] = 1;
        res[1] = 1;
        res[2] = 2;

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

        return res[n];
    }

    let n = 4;
      document.write(countWays(n));

// This code is contributed by rameshtravel07.
</script>

Output

7

• Working:

1 -> 1 -> 1 -> 1
1 -> 1 -> 2
1 -> 2 -> 1
1 -> 3
2 -> 1 -> 1
2 -> 2
3 -> 1

So Total ways: 7

• Complexity Analysis:
  • Time Complexity: O(n). 
    Only one traversal of the array is needed. So Time Complexity is O(n).
  • Space Complexity: O(n). 
    To store the values in a DP, n extra space is needed.

Method 3: Matrix Exponentiation (O(logn) Approach)

Matrix Exponentiation is mathematical ways to solve DP problem in better time complexity. Matrix Exponentiation Technique has Transformation matrix of Size K X K and Functional Vector (K X 1) .By taking n-1th power of Transformation matrix and Multiplying It With functional vector Give Resultant Vector say it Res of Size K X 1. First Element of Res will be Answer for given n value. This Approach Will Take O(K^3logn) Time Complexity Which Is Complexity of Finding (n-1) power of Transformation Matrix.

Key Terms:

K = No of Terms in which F(n) depend ,from Recurrence Relation We can Say That F(n) depend On F(n-1) and F(n-2). => K =3

F1 =  Vector (1D array) that contain F(n) value of First K terms. Since K=3 =>F1 will have F(n) value of first 2 terms. F1=[1,2,4]

T = Transformation Matrix that is a 2D matrix of Size K X K and  Consist Of All 1 After Diagonal And Rest All Zero except last row. Last Row Will have coefficient Of all K terms in which F(n)  depends In Reverse Order. => T =[ [0 1 0] ,[0 0 1], [1 1 1] ].

Algorithms:

1)Take Input N
2)If N < K then Return Precalculated Answer  //Base Condition
3)construct F1 Vector and T (Transformation Matrix)
4)Take N-1th  power of T by using  Optimal Power(T,N) Methods and assign it in T
5)return (TXF)[1]

for Optimal Power(T, N) Methods Refer Following Article: https://www.geeksforgeeks.org/write-a-c-program-to-calculate-powxn/

C++

#include <bits/stdc++.h>
#define k 3
using namespace std;

// Multiply Two Matrix Function
vector<vector<int> > multiply(vector<vector<int> > A,
                              vector<vector<int> > B)
{
    // third matrix to store multiplication of Two matrix9*
    vector<vector<int> > C(k + 1, vector<int>(k + 1));

    for (int i = 1; i <= k; i++) {
        for (int j = 1; j <= k; j++) {
            for (int x = 1; x <= k; x++) {
                C[i][j] = (C[i][j] + (A[i][x] * B[x][j]));
            }
        }
    }

    return C;
}
// Optimal Way For finding pow(t,n)
// If n Is Odd then It Will be t*pow(t,n-1)
// else return pow(t,n/2)*pow(t,n/2)
vector<vector<int> > pow(vector<vector<int> > t, int n)
{
    // base Case
    if (n == 1) {
        return t;
    }
    // Recurrence Case
    if (n & 1) {
        return multiply(t, pow(t, n - 1));
    }
    else {
        vector<vector<int> > X = pow(t, n / 2);
        return multiply(X, X);
    }
}

int compute(int n)
{
    // base Case
    if (n == 0)
        return 1;
    if (n == 1)
        return 1;
    if (n == 2)
        return 2;

    // Function Vector(indexing 1 )
    // that is [1,2]
    int f1[k + 1] = {};
    f1[1] = 1;
    f1[2] = 2;
    f1[3] = 4;

    // Constructing Transformation Matrix that will be
    /*[[0,1,0],[0,0,1],[3,2,1]]
     */
    vector<vector<int> > t(k + 1, vector<int>(k + 1));
    for (int i = 1; i <= k; i++) {
        for (int j = 1; j <= k; j++) {
            if (i < k) {
                // Store 1 in cell that is next to diagonal
                // of Matrix else Store 0 in cell
                if (j == i + 1) {
                    t[i][j] = 1;
                }
                else {
                    t[i][j] = 0;
                }
                continue;
            }
            // Last Row - store the Coefficients in reverse
            // order
            t[i][j] = 1;
        }
    }

    // Computing T^(n-1) and Setting Transformation matrix T
    // to T^(n-1)
    t = pow(t, n - 1);
    int sum = 0;
    // Computing first cell (row=1,col=1) For Resultant
    // Matrix TXF
    for (int i = 1; i <= k; i++) {
        sum += t[1][i] * f1[i];
    }
    return sum;
}
int main()
{
    int n = 4;
    cout << compute(n) << endl;
    n = 5;
    cout << compute(n) << endl;
    n = 10;
    cout << compute(n) << endl;

    return 0;
}

Java

import java.io.*;
import java.util.*;

class GFG {

    static int k = 3;

    // Multiply Two Matrix Function
    static int[][] multiply(int[][] A, int[][] B)
    {

        // Third matrix to store multiplication
        // of Two matrix9*
        int[][] C = new int[k + 1][k + 1];

        for (int i = 1; i <= k; i++) {
            for (int j = 1; j <= k; j++) {
                for (int x = 1; x <= k; x++) {
                    C[i][j]
                        = (C[i][j] + (A[i][x] * B[x][j]));
                }
            }
        }
        return C;
    }

    // Optimal Way For finding pow(t,n)
    // If n Is Odd then It Will be t*pow(t,n-1)
    // else return pow(t,n/2)*pow(t,n/2)
    static int[][] pow(int[][] t, int n)
    {

        // Base Case
        if (n == 1) {
            return t;
        }

        // Recurrence Case
        if ((n & 1) != 0) {
            return multiply(t, pow(t, n - 1));
        }
        else {
            int[][] X = pow(t, n / 2);
            return multiply(X, X);
        }
    }

    static int compute(int n)
    {

        // Base Case
        if (n == 0)
            return 1;
        if (n == 1)
            return 1;
        if (n == 2)
            return 2;

        // Function int(indexing 1 )
        // that is [1,2]
        int f1[] = new int[k + 1];
        f1[1] = 1;
        f1[2] = 2;
        f1[3] = 4;

        // Constructing Transformation Matrix that will be
        /*[[0,1,0],[0,0,1],[3,2,1]]
         */
        int[][] t = new int[k + 1][k + 1];
        for (int i = 1; i <= k; i++) {
            for (int j = 1; j <= k; j++) {
                if (i < k) {

                    // Store 1 in cell that is next to
                    // diagonal of Matrix else Store 0 in
                    // cell
                    if (j == i + 1) {
                        t[i][j] = 1;
                    }
                    else {
                        t[i][j] = 0;
                    }
                    continue;
                }

                // Last Row - store the Coefficients
                // in reverse order
                t[i][j] = 1;
            }
        }

        // Computing T^(n-1) and Setting
        // Transformation matrix T to T^(n-1)
        t = pow(t, n - 1);
        int sum = 0;

        // Computing first cell (row=1,col=1)
        // For Resultant Matrix TXF
        for (int i = 1; i <= k; i++) {
            sum += t[1][i] * f1[i];
        }
        return sum;
    }

    // Driver Code
    public static void main(String[] args)
    {

        // Input
        int n = 4;
        System.out.println(compute(n));
        n = 5;
        System.out.println(compute(n));
        n = 10;
        System.out.println(compute(n));
    }
}

// This code is contributed by Shubhamsingh10

Python3

k = 3

# Multiply Two Matrix Function


def multiply(A, B):

    # third matrix to store multiplication of Two matrix9*
    C = [[0 for x in range(k+1)] for y in range(k+1)]

    for i in range(1, k+1):
        for j in range(1, k+1):
            for x in range(1, k+1):
                C[i][j] = (C[i][j] + (A[i][x] * B[x][j]))

    return C

# Optimal Way For finding pow(t,n)
# If n Is Odd then It Will be t*pow(t,n-1)
# else return pow(t,n/2)*pow(t,n/2)


def pow(t,  n):
    # base Case
    if (n == 1):
        return t
    # Recurrence Case
    if (n & 1):
        return multiply(t, pow(t, n - 1))
    else:
        X = pow(t, n // 2)
    return multiply(X, X)


def compute(n):
    # base Case
    if (n == 0):
        return 1
    if (n == 1):
        return 1
    if (n == 2):
        return 2

    # Function Vector(indexing 1 )
    # that is [1,2]
    f1 = [0]*(k + 1)
    f1[1] = 1
    f1[2] = 2
    f1[3] = 4

    # Constructing Transformation Matrix that will be
    # [[0,1,0],[0,0,1],[3,2,1]]

    t = [[0 for x in range(k+1)] for y in range(k+1)]
    for i in range(1, k+1):
        for j in range(1, k+1):
            if (i < k):
                # Store 1 in cell that is next to diagonal of Matrix else Store 0 in
                # cell
                if (j == i + 1):
                    t[i][j] = 1
                else:
                    t[i][j] = 0
                continue
            # Last Row - store the Coefficients in reverse order
            t[i][j] = 1

    # Computing T^(n-1) and Setting Transformation matrix T to T^(n-1)
    t = pow(t, n - 1)
    sum = 0
    # Computing first cell (row=1,col=1) For Resultant Matrix TXF
    for i in range(1, k+1):
        sum += t[1][i] * f1[i]
    return sum


# Driver Code
n = 4
print(compute(n))

n = 5
print(compute(n))

n = 10
print(compute(n))

# This code is contributed by Shubhamsingh10

C#

// C# program for the above approach
using System;

class GFG {

    static int k = 3;

    // Multiply Two Matrix Function
    static int[, ] multiply(int[, ] A, int[, ] B)
    {

        // Third matrix to store multiplication
        // of Two matrix9*
        int[, ] C = new int[k + 1, k + 1];

        for (int i = 1; i <= k; i++) {
            for (int j = 1; j <= k; j++) {
                for (int x = 1; x <= k; x++) {
                    C[i, j]
                        = (C[i, j] + (A[i, x] * B[x, j]));
                }
            }
        }
        return C;
    }

    // Optimal Way For finding pow(t,n)
    // If n Is Odd then It Will be t*pow(t,n-1)
    // else return pow(t,n/2)*pow(t,n/2)
    static int[, ] pow(int[, ] t, int n)
    {

        // Base Case
        if (n == 1) {
            return t;
        }

        // Recurrence Case
        if ((n & 1) != 0) {
            return multiply(t, pow(t, n - 1));
        }
        else {
            int[, ] X = pow(t, n / 2);
            return multiply(X, X);
        }
    }

    static int compute(int n)
    {

        // Base Case
        if (n == 0)
            return 1;
        if (n == 1)
            return 1;
        if (n == 2)
            return 2;

        // Function int(indexing 1 )
        // that is [1,2]
        int[] f1 = new int[k + 1];
        f1[1] = 1;
        f1[2] = 2;
        f1[3] = 4;

        // Constructing Transformation Matrix that will be
        /*[[0,1,0],[0,0,1],[3,2,1]]
         */
        int[, ] t = new int[k + 1, k + 1];
        for (int i = 1; i <= k; i++) {
            for (int j = 1; j <= k; j++) {
                if (i < k) {

                    // Store 1 in cell that is next to
                    // diagonal of Matrix else Store 0 in
                    // cell
                    if (j == i + 1) {
                        t[i, j] = 1;
                    }
                    else {
                        t[i, j] = 0;
                    }
                    continue;
                }

                // Last Row - store the Coefficients
                // in reverse order
                t[i, j] = 1;
            }
        }

        // Computing T^(n-1) and Setting
        // Transformation matrix T to T^(n-1)
        t = pow(t, n - 1);
        int sum = 0;

        // Computing first cell (row=1,col=1)
        // For Resultant Matrix TXF
        for (int i = 1; i <= k; i++) {
            sum += t[1, i] * f1[i];
        }
        return sum;
    }

    // Driver Code
    static public void Main()
    {

        // Input
        int n = 4;
        Console.WriteLine(compute(n));
        n = 5;
        Console.WriteLine(compute(n));
        n = 10;
        Console.WriteLine(compute(n));
    }
}

// This code is contributed by Shubhamsingh10

Javascript

<script>

let k = 3;

// Multiply Two Matrix Function
function multiply(A,B)
{
    // Third matrix to store multiplication
    // of Two matrix9*
    let C = new Array(k + 1);
    for(let i=0;i<k+1;i++)
    {
        C[i]=new Array(k+1);
        for(let j=0;j<k+1;j++)
        {
            C[i][j]=0;
        }
    }

    for(let i = 1; i <= k; i++)
    {
        for(let j = 1; j <= k; j++)
        {
            for(let x = 1; x <= k; x++)
            {
                C[i][j] = (C[i][j] + (A[i][x] * B[x][j]));
            }
        }
    }
    return C;
}

// Optimal Way For finding pow(t,n)
// If n Is Odd then It Will be t*pow(t,n-1)
// else return pow(t,n/2)*pow(t,n/2)
function pow(t,n)
{
    // Base Case
    if (n == 1)
    {
        return t;
    }

    // Recurrence Case
    if ((n & 1) != 0)
    {
        return multiply(t, pow(t, n - 1));
    }
    else
    {
        let X = pow(t, n / 2);
        return multiply(X, X);
    }
}

function compute(n)
{
    // Base Case
    if (n == 0) return 1;
    if (n == 1) return 1;
    if (n == 2) return 2;

    // Function int(indexing 1 )
    // that is [1,2]
    let f1=new Array(k + 1);
    f1[1] = 1;
    f1[2] = 2;
    f1[3] = 4;

    // Constructing Transformation Matrix that will be
    /*[[0,1,0],[0,0,1],[3,2,1]]
    */
    let t = new Array(k + 1);
    for(let i=0;i<k+1;i++)
    {
        t[i]=new Array(k+1);
        for(let j=0;j<k+1;j++)
        {
            t[i][j]=0;
        }
    }

    for(let i = 1; i <= k; i++)
    {
        for(let j = 1; j <= k; j++)
        {
            if (i < k)
            {

                // Store 1 in cell that is next to
                // diagonal of Matrix else Store 0 in
                // cell
                if (j == i + 1)
                {
                    t[i][j] = 1;
                }
                else
                {
                    t[i][j] = 0;
                }
                continue;
            }

            // Last Row - store the Coefficients
            // in reverse order
            t[i][j] = 1;
        }
    }

    // Computing T^(n-1) and Setting
    // Transformation matrix T to T^(n-1)
    t = pow(t, n - 1);
    let sum = 0;

    // Computing first cell (row=1,col=1)
    // For Resultant Matrix TXF
    for(let i = 1; i <= k; i++)
    {
        sum += t[1][i] * f1[i];
    }
    return sum;
}

// Driver Code
// Input
let n = 4;
document.write(compute(n)+"<br>");
n = 5;
document.write(compute(n)+"<br>");
n = 10;
document.write(compute(n)+"<br>");


// This code is contributed by avanitrachhadiya2155

</script>

Output

7
13
274

Explanation:
We Know For This Question 
Transformation Matrix M= [[0,1,0],[0,0,1],[1,1,1]]
Functional Vector F1 = [1,2,4]
for n=2 :
    ans = (M X F1)[1]  
    ans = [2,4,7][1]  
    ans = 2 //[2,4,7][1] = First cell value of [2,4,7] i.e 2
for n=3 :
    ans = (M X M X F1)[1]  //M^(3-1) X F1 = M X M X F1
    ans = (M X [2,4,7])[1] 
    ans = [4,7,13][1]
    ans = 4
for n = 4 :
    ans = (M^(4-1) X F1)[1]
    ans = (M X M X M X F1) [1] 
    ans = (M X [4,7,13])[1] 
    ans = [7,13,24][1]
    ans = 7
for n = 5 :
    ans = (M^4 X F1)[1]
    ans = (M X [7,13,24])[1]
    ans = [13,24,44][1]
    ans = 13

Time Complexity:

O(K^3log(n)) //For Computing pow(t,n-1)
For this question K is 3
So Overall Time Complexity is O(27log(n))=O(logn)

Method 4: Using four variables

The idea is based on the Fibonacci series but here with 3 sums. we will hold the values of the first three stairs in 3 variables and will use the fourth variable to find the number of ways.

C++

// A C++ program to count number of ways
// to reach nth stair when
#include <iostream>
using namespace std;

// A recursive function used by countWays
int countWays(int n)
{
    int a = 1, b = 2, c = 4; // declaring three variables
                             // and holding the ways
                             // for first three stairs
    int d = 0; // fourth variable
    if (n == 0 || n == 1 || n == 2)
        return n;
    if (n == 3)
        return c;

    for (int i = 4; i <= n; i++) { // starting from 4 as
        d = c + b + a; // already counted for 3 stairs
        a = b;
        b = c;
        c = d;
    }
    return d;
}

// Driver program to test above functions
int main()
{
    int n = 4;
    cout << countWays(n);
    return 0;
}
// This code is contributed by Naveen Shah

Java

// A Java program to count number of ways
// to reach nth stair when
import java.io.*;

class GFG{

// A recursive function used by countWays
static int countWays(int n)
{

    // Declaring three variables
    // and holding the ways
    // for first three stairs
    int a = 1, b = 2, c = 4;

    // Fourth variable
    int d = 0;
    if (n == 0 || n == 1 || n == 2)
        return n;
    if (n == 3)
        return c;

    for(int i = 4; i <= n; i++)
    {
        // Starting from 4 as
        // already counted for 3 stairs
        d = c + b + a;
        a = b;
        b = c;
        c = d;
    }
    return d;
}

// Driver code
public static void main(String[] args)
{
    int n = 4;

    System.out.println(countWays(n));
}
}

// This code is contributed by shivanisinghss2110

Python3

# A Python program to count number of ways
# to reach nth stair when
# A recursive function used by countWays
def countWays(n):

    # declaring three variables
    # and holding the ways
    # for first three stairs
    a = 1
    b = 2
    c = 4

    d = 0 # fourth variable
    if (n == 0 or n == 1 or n == 2):
        return n
    if (n == 3):
        return c

    for i in range(4,n+1):

        # starting from 4 as
        d = c + b + a # already counted for 3 stairs
        a = b
        b = c
        c = d
    return d


# Driver program to test above functions
n = 4
print(countWays(n))

# This code is contributed by shivanisinghss2110

C#

// A C# program to count number of ways
// to reach nth stair when
using System;

class GFG{

// A recursive function used by countWays
static int countWays(int n)
{

    // Declaring three variables
    // and holding the ways
    // for first three stairs
    int a = 1, b = 2, c = 4;

    // Fourth variable
    int d = 0;
    if (n == 0 || n == 1 || n == 2)
        return n;
    if (n == 3)
        return c;

    for(int i = 4; i <= n; i++)
    {
        // Starting from 4 as
        // already counted for 3 stairs
        d = c + b + a;
        a = b;
        b = c;
        c = d;
    }
    return d;
}

// Driver code
public static void Main(String[] args)
{
    int n = 4;

    Console.Write(countWays(n));
}
}

// This code is contributed by shivanisinghss2110

Javascript

<script>
// A JavaScript program to count number of ways
// to reach nth stair when
// A recursive function used by countWays
function countWays( n)
{

    // Declaring three variables
    // and holding the ways
    // for first three stairs
    var a = 1, b = 2, c = 4;

    // Fourth variable
    var d = 0;
    if (n == 0 || n == 1 || n == 2)
        return n;
    if (n == 3)
        return c;

    for(var i = 4; i <= n; i++)
    {
        // Starting from 4 as
        // already counted for 3 stairs
        d = c + b + a;
        a = b;
        b = c;
        c = d;
    }
    return d;
}

// Driver code
    var n = 4;

    document.write(countWays(n));


// This code is contributed by shivanisinghss2110
</script>

Output

7

Time Complexity: O(n) 
Auxiliary Space: O(1).

Method 5: DP using memoization(Top down approach)

We can avoid the repeated work done in method 1(recursion) by storing the number of ways calculated so far.

We just need to store all the values in an array.

C++

// C++ Program to find n-th stair using step size
// 1 or 2 or 3.
#include <bits/stdc++.h>
using namespace std;

class GFG {
private:
    int findStepHelper(int n, vector<int>& dp)
    {
        // Base Case
        if (n == 0)
            return 1;
        else if (n < 0)
            return 0;
        // If subproblems are already calculated
        //then return it
        if (dp[n] != -1) {
            return dp[n];
        }

       // store the subproblems in the vector
        return dp[n] = findStepHelper(n - 3, dp)
                       + findStepHelper(n - 2, dp)
                       + findStepHelper(n - 1, dp);
    }

    // Returns count of ways to reach n-th stair
    // using 1 or 2 or 3 steps.
public:
    int findStep(int n)
    {
        vector<int> dp(n + 1, -1);
        return findStepHelper(n, dp);
    }
};

// Driver code
int main()
{
    GFG g;
    int n = 4;

    cout << g.findStep(n);
    return 0;
}

Python3

# Python Program to find n-th stair using step size
# 1 or 2 or 3.
class GFG:

    def findStepHelper(self, n, dp):

        # Base Case
        if (n == 0):
            return 1
        elif (n < 0):
            return 0

        # If subproblems are already calculated
        #then return it
        if (dp[n] != -1):
            return dp[n]

        # store the subproblems in the vector
        dp[n] = self.findStepHelper(n - 3, dp) + self.findStepHelper(n - 2, dp) + self.findStepHelper(n - 1, dp)

        return dp[n]

    # Returns count of ways to reach n-th stair
    # using 1 or 2 or 3 steps.
    def findStep(self, n):

        dp = [-1 for i in range(n + 1)]
        return self.findStepHelper(n, dp)

# Driver code
g = GFG()
n = 4

print(g.findStep(n))

# This code is contributed by shinjanpatra.

Javascript

<script>

// JavaScript Program to find n-th stair using step size
// 1 or 2 or 3.


class GFG {
    findStepHelper(n,dp)
    {
        // Base Case
        if (n == 0)
            return 1;
        else if (n < 0)
            return 0;

        // If subproblems are already calculated
        //then return it
        if (dp[n] != -1) {
            return dp[n];
        }

       // store the subproblems in the vector
        return dp[n] = this.findStepHelper(n - 3, dp)
                       + this.findStepHelper(n - 2, dp)
                       + this.findStepHelper(n - 1, dp);
    }

    // Returns count of ways to reach n-th stair
    // using 1 or 2 or 3 steps.

    findStep(n)
    {
        let dp = new Array(n + 1).fill(-1);
        return this.findStepHelper(n, dp);
    }
};

// Driver code

let g = new GFG();
let n = 4;

document.write(g.findStep(n));

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

Output

7

Complexity Analysis: 

• Time Complexity: O(n). Only one traversal of the array is needed. So Time Complexity is O(n).
• Space Complexity: O(n). To store the values in a DP, n extra space is needed. Also, stack space for recursion is needed which is again O(n)