Java Program to Find Sum of Fibonacci Series Numbers of First N Even Indexes

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding numbers, starting with 0 and 1. In this program, we calculate the sum of Fibonacci numbers present at even indexes from index 0 to 2 × N.

Only Fibonacci numbers at even indexes are included in the sum.
An iterative solution takes O(N) time, while a mathematical solution can reduce it to O(log N).

Illustration

Input: N = 4
Output: 33
Input: N = 7
Output: 609

Approach 1: Using Dynamic Programming

Generate Fibonacci numbers from 0 to 2 × N and add only those whose indexes are even.

Algorithm

Create a Fibonacci array.
Generate Fibonacci numbers up to index 2 × N.
Whenever the index is even, add the Fibonacci number to the sum.
Print the final sum.

Java

import java.io.*;

class geeksforgeeks {

    // Computing the value of first fibonacci series
    // and storing the sum of even indexed numbers
    static int Fib_Even_Sum(int N)
    {
        if (N <= 0)
            return 0;

        int fib[] = new int[2 * N + 1];
        fib[0] = 0;
        fib[1] = 1;

        // Initializing the sum
        int s = 0;

        // Adding remaining numbers
        for (int j = 2; j <= 2 * N; j++) {
            fib[j] = fib[j - 1] + fib[j - 2];

            // Only considering even indexes
            if (j % 2 == 0)
                s += fib[j];
        }

        return s;
    }

    // The Driver code
    public static void main(String[] args)
    {
        int N = 11;

        // Prints the sum of even-indexed numbers
        System.out.println(
            "Even sum of fibonacci series till number " + N
            + " is: " + +Fib_Even_Sum(N));
    }
}

Output

Even sum of fibonacci series till number 11 is: 28656

Explanation: The program first generates Fibonacci numbers iteratively and stores them in an array. While generating the sequence, it checks whether the current index is even. If it is, that Fibonacci number is added to the running sum. After processing all terms up to index 2 × N, the sum is displayed.

Approach 2: Using Fibonacci Identity (Optimized)

Mathematical Formula: The sum of Fibonacci numbers at even indexes is

F0+F2+F4+⋯+F2N=F2N+1−1F_0 + F_2 + F_4 + \cdots + F_{2N} = F_{2N+1} - 1F0+F2+F4+⋯+F2N=F2N+1−1

Therefore, instead of generating the complete sequence, we only need to compute F₂ₙ₊₁.