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I have a sequence, starting with $1$. You store the current sequence as a list, then duplicate it. In this copy, you invert it, turning $1$s into $0$s, and $0$s into $1$s. Then, you join it on to the first sequence.* So it goes:

$$1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1...$$

Using this, how do you get a function taking the input $n$, and outputting the $n$-th element of this sequence, without calculating every element up to that one?

I've used the recursive program for generating a square grid of these, as white and black squares, but want to use my line-based fractal generator instead. But for my purposes, I need a function that can calculate any element without knowing the ones before it.

*The function actually scans through the existing list, adding elements, to be more efficient, but this explanation's more intuitive.

Sil
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Nathan Skirrow
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  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments. – José Carlos Santos Aug 18 '18 at 11:27
  • You have glossed over the process by which the sequence was created. I suspect that if you describe that process in even minimal detail you will be able to answer your own question. – Steve B Aug 18 '18 at 11:42
  • It's a bit unclear why you call it a non-recursive sequence, since you have just described it recursively. Also this looks familiar, I think there was similar question regarding a sequence created by inverting previous digits, but I cannot find it. – Sil Aug 18 '18 at 11:42
  • Sil, I called it non-recursive because it does minor changes on greater scales, and I can't make this function work. – Nathan Skirrow Aug 18 '18 at 11:53
  • @sil, consider making this an answer. – NoChance Aug 18 '18 at 12:25
  • @NoChance Done, realized there are few more things to be said about this... – Sil Aug 18 '18 at 12:29

1 Answers1

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As mentioned in comments, this is inverted Thue–Morse sequence, see also https://oeis.org/A010060. There is a formula already mentioned in this question: A nice formula for the Thue–Morse sequence, which after slight modification yields (to match the inverted version) $$ t_n = \left(1+n-\sum_{k=1}^{n}(-1)^{\binom{n}{k}}\right) \bmod 3. $$ However this formula requires $O(n)$ calculations (not including calculation of binomial coefficients, although that is only required $\bmod 2$). So you might consider using modified version of recursive formula instead (mentioned also in the wiki pages), which really relies only on one previous sequence element, specifically \begin{align}t_0&=1\\ t_{2n}&=t_{n}\\ t_{2n+1}&=1-t_n.\end{align} This way you can compute the $t_n$ in $O(\ln n)$ steps. It is also quite simple to implement this recurrence without using recursive functions, since per your profile you are getting into Python, here is a Python 3 code snippet:

def t(n):
    r = 1
    while n > 0:
        if n & 1 == 1:
            r ^= 1
        n >>= 1
    return r

(try https://trinket.io/python/9ffdde598d). It is clear that $t_n$ is really just parity of number of $1$s in a binary representation of $n$...

Sil
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