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Find the Pell Number using C++
In the given problem, we are given an integer n we need to find Pn, i.e., the pell number in that position. Now, as we know, pell number is a part of a series given by this formula −Pn = 2*Pn-1 + Pn-2
With first two starting numbers − P0 = 0 and P1 = 1
Approach to find The Solution
Now we will solve this problem by two approaches: recursive and iterative.
Recursive Approach
In this formula, we will recursively apply the formula of Pell Number and do n iterations.
Example
#include <iostream>
using namespace std;
int pell(int n) {
if(n <= 2)
return n;
return 2*pell(n-1) + pell(n-2);
}
int main() {
int n = 6; // given n
cout << pell(n) <<"\n"; // Pell number at that position.
return 0;
}Output
70
Explanation of the above code
In this approach, we use recursion by calling pell(n-1) && pell(n-2) till n becomes less than or equal to 2 as we know the pell numbers till 2 are the same as the given number. The overall time complexity of the above program is O(N), where N is the given number.
Iterative Approach
In this approach, we will use the same formula as above but calculate the number using a for loop instead of a recursive function.
Example
#include <iostream>
using namespace std;
int main() {
int n = 6; // given n.
int p0 = 0; // initial value of pn-2.
int p1 = 1; // initial value of pn-1.
int pn; // our answer.
if(n <= 2) // if n <= 2 we print n.
cout << n <<"\n";
else {
for(int i = 2; i <= n; i++) { // we are going to find from the second number till n.
pn = 2*p1 + p0;
p0 = p1; // pn-1 becomes pn-2 for new i.
p1 = pn; // pn becomes pn-1 for new i.
}
cout << pn << "\n";
}
return 0;
}Output
70
Explanation of the above code
In the given program, we are traversing from 2 till n and simply updating the values for pn-2 to pn-1 and pn-1 to pn till we reach n.
Conclusion
In this article, we solved the problem of finding the Nth pell number using recursion and iteration. We also learned the C++ program for this problem and the complete approach ( Normal and efficient ) by which we solved this problem. We can write the same program in other languages such as C, java, python, and other languages.
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