（Java）递归函数：使用n-1和n + 1

Question

I was recently given a project as follows:

Recursive relationship : x(n) = x(n+1) - 2x(n-1) where x(0) = 1 and x(2) = 3 Write a program where the user enters a number, n, and the nth value of x, as shown above is output.

so far, I have made this method ( I am very new to, and really bad with [so far] recursion)

public class x 
{
    static int x(int n)
    {
        if(n <= 1)
        {
            return 1;
        }
        if(n == 2)
        {
            return 3;
        }
        else
        {
            return x(n+1) - 2*(x(n-1));
        }
    }

and a simple main method that prompts the user for an input of n, and then prints the result of the above method:

public static void main(String[]args)
    {
        Scanner scanner = new Scanner(System.in);
        System.out.println("Enter a number: ");
        int n = scanner.nextInt();
        System.out.println(x(n));
    }

From the assignment, I calculated that x(1) = 1. [Being I am given x(0) = 1 and x(2) = 3]:

x(n) = x(n+1) - 2(x(n-1))
x(1) = x(2) - 2(x(0))
x(1) = 3 - 2(1)
x(1) = 3 - 2
x(1) = 1

which brought me to my base case shown in the static method x above,

if(n <= 1) // because 0 and 1 each return 1
{
return 1;
}
if(n == 2) // simply because 2 returns 3
{
return 3;
}

When I run what I have, the program gives me the error :

Exception in thread "main" java.lang.StackOverflowError
    at x.x(x.java:21)
Java Result: 1

Line 21 is the return statement in the else clause of the x method:

return x(n+1) - 2*(x(n-1));

and I have no idea what I can change to fix it. I have tried restating it as x(n+1)-x(n-1)-x(n-1), but it continues to give me the same error. I don't know if it is the base case being invalid and therefore not allowing the function to run it's course properly, or just the return statement I made is wrong. I also tried writing it out as if x(5) calls x(4) calls x(3), so on as my notes show, but I arrive at an issue regarding having n+1 and n-1 in the same function statement, so I think I am just going about it the wrong way. Any guidance on this is extremely appreciated!

Answer 1

Issue is because it is never ending loop.... x(n) = x(n+1) - 2x(n-1)

You are incrementing n,ie, for

x(0) = 1

x(1) = x(2) - 2 x(0)

x(3) = x(4) - 2 x(2)....

It keeps on going as there is no upper limit for x(n+1)....

Please check if you are not missing boundary condition for it from where recursion will terminate.

Answer 2

Recurrence relation is defined as a function of the preceding terms. So you have to redefine your recursive relationship as,

x(n) = x(n+1) - 2x(n-1)

x(n+1) = x(n) + 2x(n-1)

x(n) = x(n - 1) + 2x(n-2)

Use above equation to implement the algorithm with base cases x(0) = x(1) = 1, x(2) = 3 .