Showing time limit error..please help me reduce the time conplexity

#1

#include
using namespace std;
int binarystrings(char *arr,int n,int i)
{
if(i==n)
{
arr[i]=’\0’;
return 1;
}
if(arr[0]==’\0’)
{
arr[0]=‘0’;
int ans=binarystrings(arr,n,i+1);

arr[0]=‘1’;
int ans1=binarystrings(arr,n,i+1);
return ans+ans1;
}
if(arr[i-1]!=‘1’)
{
arr[i]=‘0’;
int a=binarystrings(arr,n,i+1);
arr[i]=‘1’;
int b=binarystrings(arr,n,i+1);
return a+b;
}
else
{
arr[i]=‘0’;

int a=binarystrings(arr,n,i+1);
return a;

}

}
int main() {
int t;
cin>>t;
int q=1;
int n;
while(q<=t)
{ cin>>n;
char arr[n+1];
arr[0]=’\0’;
int ans=binarystrings(arr,n,0);
cout<<ans<<endl;
q++;

}
return 0;
}

#2

if you watch the series , you will find it as a fibonacci series with dp[1]=2 and dp[2]=3 , and dp[n]=dp[n-1] + dp[n-2] .
We can append either 0 or 1 to a string ending in 0, but we can only append 0 to a string ending in 1.
Let a[i] be the number of binary strings of length i which do not contain any two consecutive 1’s and which end in 0. Similarly, let b[i] be the number of such strings which end in 1.
this yields the above recurrence.

#3

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