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

For a given positive integer N, the purpose is to find the value of F2 + F4 + F6 +………+ F2n till N number. Where Fi indicates the i’th Fibonacci number.

The Fibonacci Series is the numbers in the below-given integer sequence.

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

Examples:

Input: n = 4
Output: 33
N = 4, So here the fibonacci series will be produced from 0th term till 8th term:
0, 1, 1, 2, 3, 5, 8, 13, 21
Sum of numbers at even indexes = 0 + 1 + 3 + 8 + 21 = 33.

Input: n = 7
Output: 609
0 + 1 + 3 + 8 + 21 + 55 + 144 + 377 = 609.

Approach 1:

Find all Fibonacci numbers till 2n and add up only the even indices.

Java

// Java Program to find even sum of
// fibonacci Series Till number N
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

Time Complexity: O(n)

Auxiliary Space: O(n) as it is using an auxiliary array fib

Approach 2:

It can be clearly seen that the required sum can be obtained thus:
2 ( F₂ + F₄ + F₆ +………+ F_2n ) = (F₁ + F₂ + F₃ + F₄ +………+ F_2n) – (F₁ – F₂ + F₃ – F₄ +………+ F_2n)

Now the first term can be obtained if we put 2n instead of n in the formula given here.

Thus F₁ + F₂ + F₃ + F₄ +………+ F_2n = F_2n+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 +………- F_2n = 1 + (-1)²ⁿ⁺¹F_2n-1 = 1 – F_2n-1.

So, 2 ( F₂ + F₄ + F₆ +………+ F_2n)
= F_2n+2 – 1 – 1 + F_2n-1
= F_2n+2 + F_2n-1 – 2
= F_2n + F_2n+1 + F_2n+1 – F_2n – 2
= 2 ( F_2n+1 -1)
Hence, ( F₂ + F₄ + F₆ +………+ F_2n) = F_2n+1 -1 .

The task is to find only F_2n+1 -1.

Below is the implementation of the above approach:

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 = 11;

        // Find the alternating sum
        System.out.println(
            "Even indexed Fibonacci Sum upto " + n
            + " terms: " + calculateEvenSum(n));
    }
}

Output

Even indexed Fibonacci Sum upto 8 terms: 1596

Time Complexity: O(log n)

Auxiliary Space: O(MAX) because using an auxiliary array