1137. N-th Tribonacci Number - Explanation

Description

The Tribonacci sequence Tn is defined as follows:

T0 = 0, T1 = 1, T2 = 1, and Tn+3 = Tn + Tn+1 + Tn+2 for n >= 0.

Given n, return the value of Tn.

Example 1:

Input: n = 3

Output: 2

Explanation:
T_3 = 0 + 1 + 1 = 2

Example 2:

Input: n = 21

Output: 121415

Constraints:

0 <= n <= 37
The answer is guaranteed to fit within a 32-bit integer, ie. answer <= 2^31 - 1.

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1. Recursion

class Solution:
    def tribonacci(self, n: int) -> int:
        if n <= 2:
            return 1 if n != 0 else 0
        return self.tribonacci(n - 1) + self.tribonacci(n - 2) + self.tribonacci(n - 3)

Time & Space Complexity

Time complexity: $O(3 ^ n)$
Space complexity: $O(n)$

2. Dynamic Programming (Top-Down)

class Solution:
    def __init__(self):
        self.dp = {}

    def tribonacci(self, n: int) -> int:
        if n <= 2:
            return 1 if n != 0 else 0
        if n in self.dp:
            return self.dp[n]

        self.dp[n] = self.tribonacci(n - 1) + self.tribonacci(n - 2) + self.tribonacci(n - 3)
        return self.dp[n]

Time & Space Complexity

Time complexity: $O(n)$
Space complexity: $O(n)$

3. Dynamic Programming (Bottom-Up)

class Solution:
    def tribonacci(self, n: int) -> int:
        if n <= 2:
            return 1 if n != 0 else 0

        dp = [0] * (n + 1)
        dp[1] = dp[2] = 1
        for i in range(3, n + 1):
            dp[i] = dp[i - 1] + dp[i - 2] + dp[i - 3]
        return dp[n]

Time & Space Complexity

Time complexity: $O(n)$
Space complexity: $O(n)$

4. Dynamic Programming (Space Optimized)

class Solution:
    def tribonacci(self, n: int) -> int:
        t = [0, 1, 1]

        if n < 3:
            return t[n]

        for i in range(3, n + 1):
            t[i % 3] = sum(t)
        return t[n % 3]

Time & Space Complexity

Time complexity: $O(n)$
Space complexity: $O(1)$