Given an integer **N**, representing the number of stations lying between the source and the destination. There are three trains available from every station and their stoppage patterns are as follows:

**Train 1:**Stops at every station**Train 2:**Stops at every alternate station**Train 3:**Stops at every third station

The task is to find the number of ways to reach the destination from the source using any possible combination of trains.

**Examples:**

Input:N = 2Output:4Explanation:

Four possible ways exists to travel from source to destination with 2 stations in between:

Train 1 (from source) -> Train 1 (from station 1) -> Train 1(from station 2) -> Destination

Train 2 (from source) -> Train 1 (from station 2) -> Destination

Train 1 (from source) -> Train 2 (from station 1) -> Destination

Train 3 (from source) -> Destination

Input:N = 0Output:1Explanation:No station is present in between the source and destination. Therefore, there is only one way to travel, i.e.

Train 1(from source) -> Destination

**Approach: **The main idea to solve the problem is to use Recursion with Memoization to solve this problem. The recurrence relation is as follows:

F(N) = F(N – 1) + F(N – 2) + F(N – 3),where,

F(N – 1)counts ways to travel upto (N – 1)^{th}station.F(N – 2)counts ways to travel upto (N – 2)^{th}station.F(N – 3)counts ways to travel upto (N – 3)^{th}station.

Follow the steps below to solve the problem:

- Initialize an array
**dp[]**for memorization. Set all indices to**-1**initially. - Define a recursive function
**findWays()**to calculate the number of ways to reach the**N**^{th}station. - Following base cases are required to be considered:
- For
**x < 0**return**0.** - For
**x = 0**, return**1.** - For
**x = 1**, return**2.** - For
**x = 2,**return**4.**

- For
- If the current state, say
**x**, is already evaluated i.e.**dp[x]**is not equal to**-1**, simply return the evaluated value. - Otherwise, calculate
**findWays(x – 1)**,**findWays(x – 2)**and**findWays(x – 3)**recursively and store their sum in**dp[x]**. - Return
**dp[x]**.

Below is the implementation of the above approach:

## C++14

`// C++ program of the above approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Dp table for memoization` `int` `dp[100000];` `// Function to count the number` `// of ways to N-th station` `int` `findWays(` `int` `x)` `{` ` ` `// Base Cases` ` ` `if` `(x < 0)` ` ` `return` `0;` ` ` `if` `(x == 0)` ` ` `return` `1;` ` ` `if` `(x == 1)` ` ` `return` `2;` ` ` `if` `(x == 2)` ` ` `return` `4;` ` ` `// If current state is` ` ` `// already evaluated` ` ` `if` `(dp[x] != -1)` ` ` `return` `dp[x];` ` ` `// Recursive calls` ` ` `// Count ways in which` ` ` `// train 1 can be chosen` ` ` `int` `count = findWays(x - 1);` ` ` `// Count ways in which` ` ` `// train 2 can be chosen` ` ` `count += findWays(x - 2);` ` ` `// Count ways in which` ` ` `// train 3 can be chosen` ` ` `count += findWays(x - 3);` ` ` `// Store the current state` ` ` `dp[x] = count;` ` ` `// Return the number of ways` ` ` `return` `dp[x];` `}` `// Driver Code` `int` `main()` `{` ` ` `// Given Input` ` ` `int` `n = 4;` ` ` `// Initialize DP table with -1` ` ` `memset` `(dp, -1, ` `sizeof` `(dp));` ` ` `// Function call to count` ` ` `// the number of ways to` ` ` `// reach the n-th station` ` ` `cout << findWays(n) << ` `"\n"` `;` `}` |

## Java

`// Java program for the above approach` `import` `java.util.*;` `class` `GFG` `{` ` ` `// Dp table for memoization` ` ` `static` `int` `dp[] = ` `new` `int` `[` `100000` `];` ` ` `// Function to count the number` ` ` `// of ways to N-th station` ` ` `static` `int` `findWays(` `int` `x)` ` ` `{` ` ` `// Base Cases` ` ` `if` `(x < ` `0` `)` ` ` `return` `0` `;` ` ` `if` `(x == ` `0` `)` ` ` `return` `1` `;` ` ` `if` `(x == ` `1` `)` ` ` `return` `2` `;` ` ` `if` `(x == ` `2` `)` ` ` `return` `4` `;` ` ` `// If current state is` ` ` `// already evaluated` ` ` `if` `(dp[x] != -` `1` `)` ` ` `return` `dp[x];` ` ` `// Recursive calls` ` ` `// Count ways in which` ` ` `// train 1 can be chosen` ` ` `int` `count = findWays(x - ` `1` `);` ` ` `// Count ways in which` ` ` `// train 2 can be chosen` ` ` `count += findWays(x - ` `2` `);` ` ` `// Count ways in which` ` ` `// train 3 can be chosen` ` ` `count += findWays(x - ` `3` `);` ` ` `// Store the current state` ` ` `dp[x] = count;` ` ` `// Return the number of ways` ` ` `return` `dp[x];` ` ` `}` ` ` `// Driven Code` ` ` `public` `static` `void` `main(String[] args)` ` ` `{` ` ` `// Given Input` ` ` `int` `n = ` `4` `;` ` ` `// Initialize DP table with -1` ` ` `Arrays.fill(dp, -` `1` `);` ` ` `// Function call to count` ` ` `// the number of ways to` ` ` `// reach the n-th station` ` ` `System.out.print(findWays(n));` ` ` `}` `}` `// This code is contributed by splevel62.` |

## Python3

`# Python3 program of the above approach` `# Dp table for memoization` `dp ` `=` `[` `-` `1` `for` `i ` `in` `range` `(` `100000` `)]` `# Function to count the number` `# of ways to N-th station` `def` `findWays(x):` ` ` ` ` `# Base Cases` ` ` `if` `(x < ` `0` `):` ` ` `return` `0` ` ` `if` `(x ` `=` `=` `0` `):` ` ` `return` `1` ` ` `if` `(x ` `=` `=` `1` `):` ` ` `return` `2` ` ` `if` `(x ` `=` `=` `2` `):` ` ` `return` `4` ` ` `# If current state is` ` ` `# already evaluated` ` ` `if` `(dp[x] !` `=` `-` `1` `):` ` ` `return` `dp[x]` ` ` `# Recursive calls` ` ` `# Count ways in which` ` ` `# train 1 can be chosen` ` ` `count ` `=` `findWays(x ` `-` `1` `)` ` ` `# Count ways in which` ` ` `# train 2 can be chosen` ` ` `count ` `+` `=` `findWays(x ` `-` `2` `)` ` ` `# Count ways in which` ` ` `# train 3 can be chosen` ` ` `count ` `+` `=` `findWays(x ` `-` `3` `)` ` ` `# Store the current state` ` ` `dp[x] ` `=` `count` ` ` `# Return the number of ways` ` ` `return` `dp[x]` `# Driver Code` `if` `__name__ ` `=` `=` `'__main__'` `:` ` ` ` ` `# Given Input` ` ` `n ` `=` `4` ` ` ` ` `# Function call to count` ` ` `# the number of ways to` ` ` `# reach the n-th station` ` ` `print` `(findWays(n))` `# This code is contributed by SURENDRA_GANGWAR` |

**Output:**

13

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

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