Check if the n-th term is odd or even in a Fibonacci like sequence

Consider a sequence a₀, a₁, …, a_n, where a_i = a_i-1 + a_i-2. Given a₀, a₁, and a positive integer n. The task is to find whether a_n is odd or even.
Note that the given sequence is like Fibonacci with the difference that the first two terms can be anything instead of 0 or 1.
Examples :

Input : a0 = 2, a1 = 4, n =3
Output : Even 
a2 = 6, a3 = 10
And a3 is even.

Input : a0 = 1, a1 = 9, n = 2
Output : Even

Method 1: The idea is to find the sequence using the array and check if nth element is even or odd.

C++

// CPP Program to check if the nth is odd or even in a
// sequence where each term is sum of previous two term
#include <bits/stdc++.h>
using namespace std;
#define MAX 100

// Return if the nth term is even or odd.
bool findNature(int a, int b, int n)
{
    int seq[MAX] = { 0 };

    seq[0] = a;
    seq[1] = b;

    for (int i = 2; i <= n; i++)
        seq[i] = seq[i - 1] + seq[i - 2];

    // Return true if odd
    return (seq[n] & 1);
}

// Driven Program
int main()
{
    int a = 2, b = 4;
    int n = 3;

    (findNature(a, b, n) ? (cout << "Odd"
                                 << " ")
                         : (cout << "Even"
                                 << " "));

    return 0;
}

Java

// Java Program to check if
// the nth is odd or even
// in a sequence where each
// term is sum of previous
// two term

// Return if the nth
// term is even or odd.
class GFG
{
public static int findNature(int a,
                             int b, int n)
{
    int[] seq = new int[100];

    seq[0] = a;
    seq[1] = b;

    for (int i = 2; i <= n; i++)
        seq[i] = seq[i - 1] + seq[i - 2];

    // Return true if odd
    if((seq[n] & 1) != 0)
    return 1;
    else
    return 0;
}

// Driver Code
public static void main(String[] args)
{
    int a = 2, b = 4;
    int n = 3;
    if(findNature(a, b, n) == 1)
    System.out.println("Odd ");
    else
    System.out.println("Even ");
}
}

// This code is contributed
// by mits

Python3

# Python3 Program to check if
# the nth is odd or even in a
# sequence where each term is
# sum of previous two term
MAX = 100;

# Return if the nth
# term is even or odd.
def findNature(a, b, n):
    seq = [0] * MAX;
    seq[0] = a;
    seq[1] = b;

    for i in range(2, n + 1):
        seq[i] = seq[i - 1] + seq[i - 2];

    # Return true if odd
    return (seq[n] & 1);

# Driver Code
a = 2;
b = 4;
n = 3;

if(findNature(a, b, n)):
    print("Odd");
else:
    print("Even");

# This code is contributed by mits

C#

// C# Program to check if the
// nth is odd or even in a
// sequence where each term
// is sum of previous two term
using System;

// Return if the nth term
// is even or odd.
class GFG
{
public static int findNature(int a,
                             int b, int n)
{
    int[] seq = new int[100];

    seq[0] = a;
    seq[1] = b;

    for (int i = 2; i <= n; i++)
        seq[i] = seq[i - 1] +
                 seq[i - 2];

    // Return true if odd
    if((seq[n] & 1)!=0)
    return 1;
    else
    return 0;
}

// Driver Code
public static void Main()
{
    int a = 2, b = 4;
    int n = 3;
    if(findNature(a, b, n) == 1)
    Console.Write("Odd ");
    else
    Console.Write("Even ");
}
}

// This code is contributed
// by mits

PHP

<?php
// PHP Program to check if the
// nth is odd or even in a
// sequence where each term is
// sum of previous two term
$MAX = 100;

// Return if the nth
// term is even or odd.
function findNature($a, $b, $n)
{
    global $MAX;
    $seq = array_fill(0,$MAX, 0);

    $seq[0] = $a;
    $seq[1] = $b;

    for ($i = 2; $i <= $n; $i++)
        $seq[$i] = $seq[$i - 1] +
                   $seq[$i - 2];

    // Return true if odd
    return ($seq[$n] & 1);
}

// Driver Code
$a = 2;
$b = 4;
$n = 3;

if(findNature($a, $b, $n))
echo "Odd";
else
echo "Even";

// This code is contributed by mits
?>

Javascript

<script>

// Javascript Program to check if the
// nth is odd or even in a
// sequence where each term is sum
// of previous two term
var MAX = 100;

// Return if the nth term is even or odd.
function findNature(a, b, n)
{
    var seq = Array(MAX).fill(0);

    seq[0] = a;
    seq[1] = b;

    for (var i = 2; i <= n; i++)
        seq[i] = seq[i - 1] + seq[i - 2];

    // Return true if odd
    return (seq[n] & 1);
}

// Driven Program
var a = 2, b = 4;
var n = 3;
(findNature(a, b, n) ? (document.write( "Odd"
                             + " "))
                     : (document.write( "Even"
                             + " ")));

</script>

Output:

Even

Method 2 (efficient) :
Observe, the nature (odd or even) of the nth term depends on the previous terms, and the nature of the previous term depends on their previous terms and finally depends on the initial value i.e a₀ and a₁. 
So, we have four possible scenarios for a₀ and a₁: 
Case 1: When a₀ an a₁ is even
In this case each of the value in the sequence will be even only.
Case 2: When a₀ an a₁ is odd
In this case, observe a₂ is even, a₃ is odd, a₄ is odd and so on. So, we can say a_i is even if i is of form 3*k – 1, else odd.
Case 3: When a₀ is even and a₁ is odd
In this case, observe a₂ is odd, a₃ is even, a₄ and a₅ is odd, a₆ is even and so on. So, we can say, a_i is even if i is multiple of 3, else odd
Case 4: When a₀ is odd and a₁ is even
In this case, observe a₂ and a₃ is odd, a₄ is even, a₅ and a₆ is odd, a₇ is even and so on. So, we can say, a_i is even if i is of the form 3*k + 1, k >= 0, else odd.
Below is the implementation of this approach:

C++

// CPP Program to check if the nth is odd or even in a
// sequence where each term is sum of previous two term
#include <bits/stdc++.h>
using namespace std;

// Return if the nth term is even or odd.
bool findNature(int a, int b, int n)
{
    if (n == 0)
        return (a & 1);

    if (n == 1)
        return (b & 1);

    // If a is even
    if (!(a & 1)) {

        // If b is even
        if (!(b & 1))
            return false;

        // If b is odd
        else
            return (n % 3 != 0);
    }

    // If a is odd
    else {
        // If b is odd
        if (!(b & 1))
            return ((n - 1) % 3 != 0);

        // If b is even
        else
            return ((n + 1) % 3 != 0);
    }
}

// Driven Program
int main()
{
    int a = 2, b = 4;
    int n = 3;

    (findNature(a, b, n) ? (cout << "Odd"
                                 << " ")
                         : (cout << "Even"
                                 << " "));

    return 0;
}

Java

// Java Program to check if the nth is odd or even in a
// sequence where each term is sum of previous two term

class GFG{
// Return if the nth term is even or odd.
static boolean findNature(int a, int b, int n)
{
    if (n == 0)
        return (a & 1)==1?true:false;

    if (n == 1)
        return (b & 1)==1?true:false;

    // If a is even
    if ((a & 1)==0) {

        // If b is even
        if ((b & 1)==0)
            return false;

        // If b is odd
        else
            return (n % 3 != 0);
    }

    // If a is odd
    else {
        // If b is odd
        if ((b & 1)==0)
            return ((n - 1) % 3 != 0);

        // If b is even
        else
            return ((n + 1) % 3 != 0);
    }
}

// Driven Program
public static void main(String[] args)
{
    int a = 2, b = 4;
    int n = 3;

    if(findNature(a, b, n))
    System.out.println("Odd");
    else
    System.out.println("Even");

}
}
// This Code is contributed by mits

Python3

# Python3 Program to check if the
# nth is odd or even in a
# sequence where each term is
# sum of previous two term

# Return if the nth
# term is even or odd.
def findNature(a, b, n):
    if (n == 0):
        return (a & 1);

    if (n == 1):
        return (b & 1);

    # If a is even
    if ((a & 1) == 0):

        # If b is even
        if ((b & 1) == 0):
            return False;

        # If b is odd
        else:
            return True if(n % 3 != 0) else False;

    # If a is odd
    else:
        # If b is odd
        if ((b & 1) == 0):
            return True if((n - 1) % 3 != 0) else False;

        # If b is even
        else:
            return True if((n + 1) % 3 != 0) else False;

# Driver Code
a = 2;
b = 4;
n = 3;

if (findNature(a, b, n) == True):
    print("Odd", end = " ");
else:
    print("Even", end = " ");

# This code is contributed by mits

C#

// C# Program to check if the nth is odd or even in a
// sequence where each term is sum of previous two term

class GFG{
// Return if the nth term is even or odd.
static bool findNature(int a, int b, int n)
{
    if (n == 0)
        return (a & 1)==1?true:false;

    if (n == 1)
        return (b & 1)==1?true:false;

    // If a is even
    if ((a & 1)==0) {

        // If b is even
        if ((b & 1)==0)
            return false;

        // If b is odd
        else
            return (n % 3 != 0);
    }

    // If a is odd
    else {
        // If b is odd
        if ((b & 1)==0)
            return ((n - 1) % 3 != 0);

        // If b is even
        else
            return ((n + 1) % 3 != 0);
    }
}

// Driven Program
static void Main()
{
    int a = 2, b = 4;
    int n = 3;

    if(findNature(a, b, n))
    System.Console.WriteLine("Odd");
    else
    System.Console.WriteLine("Even");

}
}
// This Code is contributed by mits

PHP

<?php
// PHP Program to check if the
// nth is odd or even in a
// sequence where each term is
// sum of previous two term

// Return if the nth
// term is even or odd.
function findNature($a, $b, $n)
{
    if ($n == 0)
        return ($a & 1);

    if ($n == 1)
        return ($b & 1);

    // If a is even
    if (!($a & 1))
    {

        // If b is even
        if (!($b & 1))
            return false;

        // If b is odd
        else
            return ($n % 3 != 0);
    }

    // If a is odd
    else
    {
        // If b is odd
        if (!($b & 1))
            return (($n - 1) % 3 != 0);

        // If b is even
        else
            return (($n + 1) % 3 != 0);
    }
}

// Driver Code
$a = 2; $b = 4;
$n = 3;

if (findNature($a, $b, $n) == true)
    echo "Odd"," ";
else
    echo "Even"," ";

// This code is contributed by ajit
?>

Javascript

<script>
    // Javascript Program to check if the nth is odd or even in a
    // sequence where each term is sum of previous two term

    // Return if the nth term is even or odd.
    function findNature(a, b, n)
    {
        if (n == 0)
            return (a & 1)==1?true:false;

        if (n == 1)
            return (b & 1)==1?true:false;

        // If a is even
        if ((a & 1)==0) {

            // If b is even
            if ((b & 1)==0)
                return false;

            // If b is odd
            else
                return (n % 3 != 0);
        }

        // If a is odd
        else {
            // If b is odd
            if ((b & 1)==0)
                return ((n - 1) % 3 != 0);

            // If b is even
            else
                return ((n + 1) % 3 != 0);
        }
    }

    let a = 2, b = 4;
    let n = 3;

    if(findNature(a, b, n))
        document.write("Odd");
    else
        document.write("Even");

// This code is contributed by suresh07.
</script>

Output :

Even

Time Complexity: O(1)

Auxiliary Space: O(1)