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Sum of Fibonacci numbers at even indexes upto N terms
  • Last Updated : 01 Apr, 2021

Given a positive integer N, the task is to find the value of F2 + F4 + F6 +………+ F2n upto N terms where Fi denotes the i-th Fibonacci number.
The Fibonacci numbers are the numbers in the following integer sequence.
 

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ……

Examples: 
 

Input: n = 5 
Output: 88 
N = 5, So the fibonacci series will be generated from 0th term upto 10th term: 
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 
Sum of elements at even indexes = 0 + 1 + 3 + 8 + 21 + 55
Input: n = 8 
Output: 1596 
0 + 1 + 3 + 8 + 21 + 55 + 144 + 377 + 987 = 1596.

 



Method-1: This method includes solving the problem directly by finding all Fibonacci numbers till 2n and adding up the only the even indices. But this will require O(n) time complexity.
Below is the implementation of the above approach: 
 

C++




// C++ Program to find  sum
// of even-indiced Fibonacci numbers
#include <bits/stdc++.h>
using namespace std;
 
// Computes value of first fibonacci numbers
// and stores the even-indexed sum
int calculateEvenSum(int n)
{
    if (n <= 0)
        return 0;
 
    int fibo[2 * n + 1];
    fibo[0] = 0, fibo[1] = 1;
 
    // Initialize result
    int sum = 0;
 
    // Add remaining terms
    for (int i = 2; i <= 2 * n; i++) {
        fibo[i] = fibo[i - 1] + fibo[i - 2];
 
        // For even indices
        if (i % 2 == 0)
            sum += fibo[i];
    }
 
    // Return the alternting sum
    return sum;
}
 
// Driver program to test above function
int main()
{
 
    // Get n
    int n = 8;
 
    // Find the even-indiced sum
    cout << "Even indexed Fibonacci Sum upto "
         << n << " terms: "
         << calculateEvenSum(n) << endl;
 
    return 0;
}

Java




// Java Program to find sum
// of even-indiced Fibonacci numbers
 
import java.io.*;
 
class GFG {
 
 
// Computes value of first fibonacci numbers
// and stores the even-indexed sum
static int calculateEvenSum(int n)
{
    if (n <= 0)
        return 0;
 
    int fibo[] = new int[2 * n + 1];
    fibo[0] = 0; fibo[1] = 1;
 
    // Initialize result
    int sum = 0;
 
    // Add remaining terms
    for (int i = 2; i <= 2 * n; i++) {
        fibo[i] = fibo[i - 1] + fibo[i - 2];
 
        // For even indices
        if (i % 2 == 0)
            sum += fibo[i];
    }
 
    // Return the alternting sum
    return sum;
}
 
// Driver program
    public static void main (String[] args) {
            // Get n
    int n = 8;
 
    // Find the even-indiced sum
    System.out.println("Even indexed Fibonacci Sum upto "
        + n + " terms: "+
        + calculateEvenSum(n));
 
    }
}
 
// This code is contributed
// by shs

Python 3




# Python3 Program to find sum
# of even-indiced Fibonacci numbers
 
# Computes value of first fibonacci
# numbers and stores the even-indexed sum
def calculateEvenSum(n) :
 
    if n <= 0 :
        return 0
 
    fibo = [0] * (2 * n + 1)
    fibo[0] , fibo[1] = 0 , 1
 
    # Initialize result
    sum = 0
 
    # Add remaining terms
    for i in range(2, 2 * n + 1) :
 
        fibo[i] = fibo[i - 1] + fibo[i - 2]
 
        # For even indices
        if i % 2 == 0 :
            sum += fibo[i]
 
    # Return the alternting sum
    return sum
 
# Driver code
if __name__ == "__main__" :
 
    # Get n
    n = 8
 
    # Find the even-indiced sum
    print("Even indexed Fibonacci Sum upto",
           n, "terms:", calculateEvenSum(n))
 
# This code is contributed
# by ANKITRAI1

C#




// C# Program to find sum of
// even-indiced Fibonacci numbers
using System;
 
class GFG
{
     
// Computes value of first fibonacci
// numbers and stores the even-indexed sum
static int calculateEvenSum(int n)
{
    if (n <= 0)
        return 0;
 
    int []fibo = new int[2 * n + 1];
    fibo[0] = 0; fibo[1] = 1;
 
    // Initialize result
    int sum = 0;
 
    // Add remaining terms
    for (int i = 2; i <= 2 * n; i++)
    {
        fibo[i] = fibo[i - 1] +
                  fibo[i - 2];
 
        // For even indices
        if (i % 2 == 0)
            sum += fibo[i];
    }
 
    // Return the alternting sum
    return sum;
}
 
// Driver Code
static public void Main ()
{
    // Get n
    int n = 8;
     
    // Find the even-indiced sum
    Console.WriteLine("Even indexed Fibonacci Sum upto " +
                    n + " terms: " + calculateEvenSum(n));
}
}
 
// This code is contributed
// by Sach_Code

PHP




<?php
// PHP Program to find sum of
// even-indiced Fibonacci numbers
 
// Computes value of first fibonacci
// numbers and stores the even-indexed sum
function calculateEvenSum($n)
{
    if ($n <= 0)
        return 0;
 
    $fibo[2 * $n + 1] = array();
    $fibo[0] = 0; $fibo[1] = 1;
 
    // Initialize result
    $sum = 0;
 
    // Add remaining terms
    for ($i = 2; $i <= 2 * $n; $i++)
    {
        $fibo[$i] = $fibo[$i - 1] +
                    $fibo[$i - 2];
 
        // For even indices
        if ($i % 2 == 0)
            $sum += $fibo[$i];
    }
 
    // Return the alternting sum
    return $sum;
}
 
// Driver Code
 
// Get n
$n = 8;
 
// Find the even-indiced sum
echo "Even indexed Fibonacci Sum upto " . $n .
     " terms: " . calculateEvenSum($n) . "\n";
 
// This code is contributed
// by Akanksha Rai(Abby_akku)
?>

Javascript




<script>
// Javascript Program to find sum
// of even-indiced Fibonacci numbers
 
    // Computes value of first fibonacci numbers
    // and stores the even-indexed sum
    function calculateEvenSum( n)
    {
        if (n <= 0)
            return 0;
 
        let fibo = Array(2 * n + 1);
        fibo[0] = 0;
        fibo[1] = 1;
 
        // Initialize result
        let sum = 0;
 
        // Add remaining terms
        for ( i = 2; i <= 2 * n; i++)
        {
            fibo[i] = fibo[i - 1] + fibo[i - 2];
 
            // For even indices
            if (i % 2 == 0)
                sum += fibo[i];
        }
 
        // Return the alternting sum
        return sum;
    }
 
    // Driver program
      
        // Get n
        let n = 8;
 
        // Find the even-indiced sum
        document.write("Even indexed Fibonacci Sum upto " + n + " terms: " + +calculateEvenSum(n));
 
// This code is contributed by 29AjayKumar 
</script>
Output: 
Even indexed Fibonacci Sum upto 8 terms: 1596

 

Method-2: 
 

It can be clearly seen that the required sum can be obtained thus: 
2 ( F2 + F4 + F6 +………+ F2n ) = (F1 + F2 + F3 + F4 +………+ F2n) – (F1 – F2 + F3 – F4 +………+ F2n)
Now the first term can be obtained if we put 2n instead of n in the formula given here.
Thus F1 + F2 + F3 + F4 +………+ F2n = F2n+2 – 1.
The second term can also be found if we put 2n instead of n in the formula given here
Thus, F1 – F2 + F3 – F4 +………- F2n = 1 + (-1)2n+1F2n-1 = 1 – F2n-1.
So, 2 ( F2 + F4 + F6 +………+ F2n
= F2n+2 – 1 – 1 + F2n-1 
= F2n+2 + F2n-1 – 2 
= F2n + F2n+1 + F2n+1 – F2n – 2 
= 2 ( F2n+1 -1) 
Hence, ( F2 + F4 + F6 +………+ F2n) = F2n+1 -1 .

So in order to find the required sum, the task is to find only F2n+1 which requires O(log n) time.( Refer to method 5 or method 6 in this article.
Below is the implementation of the above approach: 
 

C++




// C++ Program to find even indexed Fibonacci Sum in
// O(Log n) time.
 
#include <bits/stdc++.h>
using namespace std;
 
const int MAX = 1000;
 
// Create an array for memoization
int f[MAX] = { 0 };
 
// Returns n'th Fibonacci number
// using table f[]
int fib(int n)
{
    // Base cases
    if (n == 0)
        return 0;
    if (n == 1 || n == 2)
        return (f[n] = 1);
 
    // If fib(n) is already computed
    if (f[n])
        return f[n];
 
    int k = (n & 1) ? (n + 1) / 2 : n / 2;
 
    // Applying above formula [Note value n&1 is 1
    // if n is odd, else 0].
    f[n] = (n & 1) ? (fib(k) * fib(k) + fib(k - 1) * fib(k - 1))
                   : (2 * fib(k - 1) + fib(k)) * fib(k);
 
    return f[n];
}
 
// Computes value of even-indexed  Fibonacci Sum
int calculateEvenSum(int n)
{
    return (fib(2 * n + 1) - 1);
}
 
// Driver program to test above function
int main()
{
    // Get n
    int n = 8;
 
    // Find the alternating sum
    cout << "Even indexed Fibonacci Sum upto "
         << n << " terms: "
         << calculateEvenSum(n) << endl;
 
    return 0;
}

Java




// Java Program to find even indexed Fibonacci Sum in
// O(Log n) time.
 
class GFG {
 
    static int MAX = 1000;
 
   // Create an array for memoization
    static int f[] = new int[MAX];
 
// Returns n'th Fibonacci number
// using table f[]
    static int fib(int n) {
        // Base cases
        if (n == 0) {
            return 0;
        }
        if (n == 1 || n == 2) {
            return (f[n] = 1);
        }
 
        // If fib(n) is already computed
        if (f[n] == 1) {
            return f[n];
        }
 
        int k = (n % 2 == 1) ? (n + 1) / 2 : n / 2;
 
        // Applying above formula [Note value n&1 is 1
        // if n is odd, else 0].
        f[n] = (n % 2 == 1) ? (fib(k) * fib(k) + fib(k - 1) * fib(k - 1))
                : (2 * fib(k - 1) + fib(k)) * fib(k);
 
        return f[n];
    }
 
// Computes value of even-indexed Fibonacci Sum
    static int calculateEvenSum(int n) {
        return (fib(2 * n + 1) - 1);
    }
 
// Driver program to test above function
    public static void main(String[] args) {
        // Get n
        int n = 8;
 
        // Find the alternating sum
        System.out.println("Even indexed Fibonacci Sum upto "
                + n + " terms: "
                + calculateEvenSum(n));
    }
}
// This code is contributed by Rajput-Ji

Python3




# Python3 Program to find even indexed
# Fibonacci Sum in O(Log n) time.
MAX = 1000;
 
# Create an array for memoization
f = [0] * MAX;
 
# Returns n'th Fibonacci number
# using table f[]
def fib(n):
     
    # Base cases
    if (n == 0):
        return 0;
    if (n == 1 or n == 2):
        f[n] = 1;
        return f[n];
 
    # If fib(n) is already computed
    if (f[n]):
        return f[n];
 
    k = (n + 1) // 2 if (n % 2 == 1) else n // 2;
 
    # Applying above formula [Note value n&1 is 1
    # if n is odd, else 0].
    f[n] = (fib(k) * fib(k) + fib(k - 1) * fib(k - 1)) \
    if (n % 2 == 1) else (2 * fib(k - 1) + fib(k)) * fib(k);
 
    return f[n];
 
# Computes value of even-indexed Fibonacci Sum
def calculateEvenSum(n):
    return (fib(2 * n + 1) - 1);
 
# Driver Code
if __name__ == '__main__':
     
    # Get n
    n = 8;
 
    # Find the alternating sum
    print("Even indexed Fibonacci Sum upto",
          n, "terms:", calculateEvenSum(n));
 
# This code is contributed by PrinciRaj1992

C#




// C# Program to find even indexed Fibonacci Sum in
// O(Log n) time.
using System;
 
class GFG
{
 
    static int MAX = 1000;
 
    // Create an array for memoization
    static int []f = new int[MAX];
 
    // Returns n'th Fibonacci number
    // using table f[]
    static int fib(int n)
    {
        // Base cases
        if (n == 0)
        {
            return 0;
        }
        if (n == 1 || n == 2)
        {
            return (f[n] = 1);
        }
 
        // If fib(n) is already computed
        if (f[n] == 1)
        {
            return f[n];
        }
 
        int k = (n % 2 == 1) ? (n + 1) / 2 : n / 2;
 
        // Applying above formula [Note value n&1 is 1
        // if n is odd, else 0].
        f[n] = (n % 2 == 1) ? (fib(k) * fib(k) +
                                fib(k - 1) * fib(k - 1))
                : (2 * fib(k - 1) + fib(k)) * fib(k);
 
        return f[n];
    }
 
    // Computes value of even-indexed Fibonacci Sum
    static int calculateEvenSum(int n)
    {
        return (fib(2 * n + 1) - 1);
    }
 
    // Driver code
    public static void Main()
    {
        // Get n
        int n = 8;
 
        // Find the alternating sum
        Console.WriteLine("Even indexed Fibonacci Sum upto "
                + n + " terms: "
                + calculateEvenSum(n));
    }
}
 
//This code is contributed by 29AjayKumar

Javascript




<script>
// javascript Program to find even indexed Fibonacci Sum in
// O(Log n) time.
    var MAX = 1000;
 
    // Create an array for memoization
    var f = Array(MAX).fill(0);
 
    // Returns n'th Fibonacci number
    // using table f
    function fib(n) {
        // Base cases
        if (n == 0) {
            return 0;
        }
        if (n == 1 || n == 2) {
            return (f[n] = 1);
        }
 
        // If fib(n) is already computed
        if (f[n] == 1) {
            return f[n];
        }
 
        var k = (n % 2 == 1) ? (n + 1) / 2 : n / 2;
 
        // Applying above formula [Note value n&1 is 1
        // if n is odd, else 0].
        f[n] = (n % 2 == 1) ? (fib(k) * fib(k) + fib(k - 1) * fib(k - 1)) : (2 * fib(k - 1) + fib(k)) * fib(k);
 
        return f[n];
    }
 
    // Computes value of even-indexed Fibonacci Sum
    function calculateEvenSum(n) {
        return (fib(2 * n + 1) - 1);
    }
 
    // Driver program to test above function
     
        // Get n
        var n = 8;
 
        // Find the alternating sum
        document.write("Even indexed Fibonacci Sum upto " + n + " terms: " + calculateEvenSum(n));
 
// This code is contributed by todaysgaurav
</script>
Output: 
Even indexed Fibonacci Sum upto 8 terms: 1596

 

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