## Solution

#### Approach 1: Brute Force with Set

Intuition

Every Fibonacci-like subsequence has each two adjacent terms determine the next expected term. For example, with 2, 5, we expect that the sequence must continue 7, 12, 19, 31, etc.

We can use a Set structure to determine quickly whether the next term is in the array A or not. Because of the exponential growth of these terms, there are at most 43 terms in any Fibonacci-like subsequence that has maximum value .

Algorithm

For each starting pair A[i], A[j], we maintain the next expected value y = A[i] + A[j] and the previously seen largest value x = A[j]. If y is in the array, then we can then update these values (x, y) -> (y, x+y).

Also, because subsequences are only fibonacci-like if they have length 3 or more, we must perform the check ans >= 3 ? ans : 0 at the end.

Complexity Analysis

• Time Complexity: , where is the length of A, and is the maximum value of A.

• Space Complexity: , the space used by the set S.

#### Approach 2: Dynamic Programming

Intuition

Think of two consecutive terms A[i], A[j] in a fibonacci-like subsequence as a single node (i, j), and the entire subsequence is a path between these consecutive nodes. For example, with the fibonacci-like subsequence (A = 2, A = 3, A = 5, A = 8, A = 13), we have the path between nodes (1, 2) <-> (2, 4) <-> (4, 7) <-> (7, 10).

The motivation for this is that two nodes (i, j) and (j, k) are connected if and only if A[i] + A[j] == A[k], and we needed this amount of information to know about this connection. Now we have a problem similar to Longest Increasing Subsequence.

Algorithm

Let longest[i, j] be the longest path ending in [i, j]. Then longest[j, k] = longest[i, j] + 1 if (i, j) and (j, k) are connected. Since i is uniquely determined as A.index(A[k] - A[j]), this is efficient: we check for each j < k what i is potentially, and update longest[j, k] accordingly.

Complexity Analysis

• Time Complexity: , where is the length of A.

• Space Complexity: , where is the largest element of A. We can show that the number of elements in a subsequence is bounded by where is the minimum element in the subsequence.

Analysis written by: @awice.