Number of sequences which has HEAD at alternate positions to the right of the first HEAD

Given that a coin is tossed N times. The task is to find the total number of the sequence of tosses such that after the first head from left, all the alternating positions to the right of it are occupied by the head only. The positions except the alternating position can be occupied by any of head or tail.

For example, if you are tossing a coin 10 times (N = 10) and the first head occurs at the 3rd position than all the further alternating positions to the right are 5, 7, 9…

Examples:

Input: N = 2
Output: 4
All possible combinations will be TT, TH, HT, HH.

Input: N = 3
Output: 6
All possible combination will be TTT, TTH, THT, THH, HTH, HHH.
In this case, HHT & HTT is not possible because in this combination
condition of alternate heads does not satisfy. Hence answer will be 6.

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:
If the sequence is to start with a tail, then all possible sequences of length N-1 is valid. Now, if the sequence is to start with a head, then every odd index in the sequence (assuming the sequence is 1-based) will be head and rest of the N/2 places can be either head or tail. Thus the recursive formula for above problem will be:

f(N) = f(N-1) + 2floor(N/2)

Mathematical Calculation :

Let fo(N) and fe(N) be the functions
that are defined for the odd and even values of N respectively.

fo(N) =  fe(N-1) + 2(N-1)/2
fe(N) =  fo(N-1) + 2N/2

From above equation compute
fo(N) = fo(N-2) + 2(N-1)/2 + 2(N-1)/2
fe(N) = fe(N-2) + 2N/2 + 2(N-2)/2

Base Case:
fo(1) = 2
fe(0) = 1

By using the above equation, compute the following results :
fo(N) - fo(N-2) =  2(N-1)/2 + 2(N-1)/2
fo(N) - fo(N-2) =  2(N+1)/2

By taking the sum of above equation for all odd values of N,
below thing is computed.
fo(N) - fo(N-2) + fo(N-1) - fo(N-3) + ...... + fo(3) - fo(1) =
22 + 23 + 24 + ..... + 2(N+1)/2

Hence on summation,
fo(N) - fo(1) = SUM[ n = 0 to (N+1)/2 ] 2n - 21 - 20

By using sum of geometric progression
fo(N) = 2( N + 1 ) / 2 + 2( N + 1 ) / 2 - 2

Similarly, find fe(N) :
fe(N) = fe(N-2) + 2N/2 + 2(N-2)/2
fe(N) - fe(0) = SUM[ n = 0 to N/2 ] 2n - 20 - SUM[ n = 0 to (N-1)/2 ] 2n

By using sum of geometric progression
fe(N) = 2 (N / 2) + 1 + 2 N / 2 - 2

The final formula will be as follows:

f(N) = 2(N+1)/2 + 2(N+1)/2 – 2, if N is odd

f(N) = 2(N/2) + 1 + 2N/2 – 2, if N is even

Below is the implementation of the above approach:

C++

 // C++ program to find number of sequences #include using namespace std;    // function to calculate total sequences possible int findAllSequence(int N) {        // Value of N is even     if (N % 2 == 0) {         return pow(2, N / 2 + 1) + pow(2, N / 2) - 2;     }     // Value of N is odd     else {         return pow(2, (N + 1) / 2) + pow(2, (N + 1) / 2) - 2;     } }    // Driver code int main() {     int N = 2;     cout << findAllSequence(N) << endl;     return 0; }

Java

 // Java program to find  // number of sequences import java.io.*;    class GFG  {    // function to calculate  // total sequences possible static int findAllSequence(int N) {        // Value of N is even     if (N % 2 == 0)      {         return (int)(Math.pow(2, N / 2 + 1) +                       Math.pow(2, N / 2) - 2);     }            // Value of N is odd     else      {         return (int)(Math.pow(2, (N + 1) / 2) +                       Math.pow(2, (N + 1) / 2) - 2);     } }    // Driver code public static void main (String[] args)  {     int N = 2;     System.out.print( findAllSequence(N)); } }    // This code is contributed // by anuj_67.

Python3

 # Python3 program to find number  # of sequences    # function to calculate total # sequences possible def findAllSequence(N):        # Value of N is even     if (N % 2 == 0):         return (pow(2, N / 2 + 1) +                  pow(2, N / 2) - 2);     # Value of N is odd     else:         return (pow(2, (N + 1) / 2) +                  pow(2, (N + 1) / 2) - 2);    # Driver code N = 2; print(int(findAllSequence(N)));    # This code is contributed by mits

C#

 // C# program to find  // number of sequences using System;  public class GFG{            // function to calculate      // total sequences possible     static int findAllSequence(int N)     {            // Value of N is even         if (N % 2 == 0)          {             return (int)(Math.Pow(2, N / 2 + 1) +                           Math.Pow(2, N / 2) - 2);         }            // Value of N is odd         else         {             return (int)(Math.Pow(2, (N + 1) / 2) +                           Math.Pow(2, (N + 1) / 2) - 2);         }     }        // Driver code     public static void Main ()      {         int N = 2;         Console.WriteLine( findAllSequence(N));     } }    // This code is contributed by 29AjayKumar

PHP



Output:

4

Time Complexity: O(LogN)

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