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Asked: 2024-09-26T04:51:57+05:30 In:

How can I efficiently compute the nth term of the Van Eck sequence in Python?

I’ve been diving into some interesting number sequences lately and stumbled upon the Van Eck sequence, which seems to have quite a unique pattern. For those unfamiliar, the sequence starts with a zero and grows based on past values in a really intriguing way. The next term is determined by how many steps back the last occurrence of the current term was. If the term hasn’t appeared before, it’s just a zero.

Here’s where I need help: I want to find the nth term of the Van Eck sequence, but I’m kind of struggling with how to implement it efficiently. I’ve seen some approaches, but they all seem a bit cumbersome. I get that when you want the nth term, you basically keep track of the numbers and their latest occurrences.

Here’s what I’ve confirmed for the starting sequence:

– The sequence begins with 0.
– The next terms follow based on where the last occurrence of the current term was found.
– For example, the first few terms are: 0, 0, 1, 0, 2, 1, 0, 3, 2, 0, 4, 3, and so on.

So, it seems like there’s a bit of a recursive pattern and a clever way to store past values. I’ve tried coding it up in Python, but the performance drops significantly when I try to calculate larger values of n (like n = 1000 or more).

Has anyone come up with a neat way to efficiently compute the nth term without bogging down the system? Any tips on algorithms, or perhaps a different perspective that might help optimize this? I’m curious to see how others might tackle this problem.

Thanks in advance for any insights or solutions! I’ve genuinely found this sequence fascinating, and I’m excited to learn more about it from whoever has gotten their hands dirty in the code as well!

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    2 Answers


    1. The Van Eck sequence can be efficiently computed by utilizing a dictionary to track the last occurrences of each number in the sequence. This allows us to quickly determine how far back to look for the last occurrence of the current term, thus avoiding the performance pitfalls associated with iterating through a list for larger values of n. Here’s a Python implementation that illustrates this approach:

      def van_eck_sequence(n):
    sequence = [0]  # Start the sequence with the first term
    last_occurrence = {0: 0}  # Dictionary to hold the last occurrence of each term

    for i in range(1, n):
        last_term = sequence[-1]  # Get the last term added to the sequence
        if last_term in last_occurrence:
            next_term = i - 1 - last_occurrence[last_term]  # Calculate the next term
        else:
            next_term = 0  # If the term hasn't appeared before, it's 0

        sequence.append(next_term)
        last_occurrence[last_term] = i - 1  # Update the last occurrence of the last term

    return sequence[-1]  # Return the nth term

# Example usage:
n = 1000
print(f"The {n}th term of the Van Eck sequence is {van_eck_sequence(n)}")

      This program efficiently computes the nth term of the Van Eck sequence using a list for the sequence and a dictionary for tracking last occurrences, ensuring that it maintains optimal performance even for larger values of n.




    2. Van Eck Sequence Calculator

      Van Eck Sequence Calculator

      If you’re looking to compute the nth term of the Van Eck sequence efficiently, here’s a simple Python implementation that should help you out!

              
def van_eck(n):
    # Dictionary to store the last occurrences of numbers
    last_occurrence = {}
    # Starting sequence
    sequence = [0]  # First term is always 0

    for i in range(1, n):
        last_term = sequence[i - 1]
        # Determine the next term based on last occurrence
        if last_term in last_occurrence:
            next_term = (i - 1) - last_occurrence[last_term]
        else:
            next_term = 0
            
        sequence.append(next_term)
        # Update the last occurrence of the last term
        last_occurrence[last_term] = i - 1

    return sequence[n - 1]

# Example: Calculate the 1000th term
nth_term = 1000
print("The {}th term of the Van Eck sequence is: {}".format(nth_term, van_eck(nth_term)))

      This code keeps track of the last occurrences of each number using a dictionary, which makes it efficient. You can change the value of nth_term to compute any term you want.

      Hope this helps spark some ideas for your own implementation!


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