How to change this nested recursion to linear recursion?

Question

I want to change this function of nested recursion to linear recursion or optimally is tail recursion.

long long x(int n) {
    if (n == 0) return 1;
    int ret = 0;
    for (int i = 0; i < n; ++i) {
        ret += (n - i) * (n - i) * x(i);
    }
    return ret;
}

the origin recursion is

X(0)=1; X(n)=n^2*X(0)+(n-1)^2*X1+...+1^2*X(n-1)

@Someprogrammerdude i implement a recursion function to solve the X(n) — HCMUSer
– HCMUSer, Commented Apr 8, 2023 at 8:01
@Someprogrammerdude the origin recursion is X(0)=1; X(n)=n^2*X(0)+(n-1)^2*X1+...+1^2*X(n-1) — HCMUSer
– HCMUSer, Commented Apr 8, 2023 at 8:02
But apparently the solution you wrote have a problem, that you want to solve. What is the problem with your solution? Why do you want to rewrite it? Please edit your question to give us more details. — Some programmer dude
– Some programmer dude, Commented Apr 8, 2023 at 8:14
@Someprogrammerdude i want to improve my solution to optimize the time and space. So how does it be implemented which not use loop? — HCMUSer
– HCMUSer, Commented Apr 8, 2023 at 8:24

SamHoque · Accepted Answer · 2023-04-08 08:31:56Z

2

Here you go, converted to tail recursion by using an accumulator variable to store the intermediate results of the computation

long long x(int n, int i, long long acc) {
    if (i == n) return acc;
    return x(n, i + 1, acc + (n - i) * (n - i) * x(i));
}

long long x(int n) {
    return x(n, 0, 1);
}

answered Apr 8, 2023 at 8:31

SamHoque

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