ABA Sequence Generation

The ABA problem involves generating a sequence of numbers where each number is the sum of the previous two numbers, but with a specific constraint: the sequence must contain a given target number. This challenge tests your ability to work with sequences, loops, and conditional logic to find a solution within defined constraints. It's a common pattern in algorithmic problem-solving, applicable to various scenarios like financial modeling or pattern recognition.

Problem Description

You are given two integers, start and target. Your task is to generate an ABA sequence starting with start as the first number. The sequence is generated by adding the previous two numbers to get the next number. The sequence must contain the target number. Return the shortest ABA sequence that includes the target. If no such sequence exists within the given constraints, return an empty list.

Key Requirements:

• The sequence must start with the given start value.
• Each subsequent number in the sequence is the sum of the previous two numbers.
• The sequence must contain the target value.
• The sequence should be as short as possible.
• If the target is equal to start, return [start].

Expected Behavior:

The function should return a list of integers representing the ABA sequence. The sequence should be generated until the target is found or the maximum length of the sequence is reached.

Edge Cases to Consider:

• start is negative.
• target is negative.
• target is zero.
• start is zero.
• target is smaller than start.
• No sequence exists within the maximum length constraint.

Examples

Example 1:

Input: start = 2, target = 10
Output: [2, 4, 6, 10]
Explanation: The sequence starts with 2. The next numbers are 2+2=4, 4+2=6, and 6+4=10. The target 10 is found, and the sequence is returned.

Example 2:

Input: start = 3, target = 5
Output: [3, 5]
Explanation: The sequence starts with 3. The next number is 3+0=3. The next number is 3+3=6. The next number is 5. The target 5 is found, and the sequence is returned.

Example 3:

Input: start = 1, target = 7
Output: [1, 2, 3, 5, 8]
Explanation: The sequence starts with 1. The next numbers are 1+0=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8. The target 7 is not found within the maximum length.

Constraints

• 1 <= start <= 100
• 1 <= target <= 1000
• The maximum length of the sequence should not exceed 20.
• Both start and target are positive integers.

Notes

Consider using a while loop to generate the sequence. Keep track of the sequence length to avoid infinite loops. Think about how to efficiently check if the target has been found. The sequence is defined by adding the previous two numbers, so you'll need to store at least the last two numbers to calculate the next one. If the target is equal to the start, the shortest sequence is simply [start].