Problem 617

[mctrafik]

D(100)=21 means there are 21 unique sequences for e, n? So like Is 2 4 2 4 2 ... different from 4 2 4 2 4 ... ?

Don't want to give too much away, but if I understand correctly, trying different e, n, and a_0 (at 2... for all) only produces D(100) = 19 so I feel like there's something I'm missing from the statement about "some e" for C(n). I mean there's a reason why D(N) isn't defined as unique MPS sequences for n <= N.

[v6ph1]

They all have a different beginning.
You may try brute force all n, a_0 and e (2...6) smaller than 100.

[traxex]

The problem statement contradicts itself and I'm surprised nobody created a thread here until now. Although the correct interpretation can be puzzled out using the provided sample values, that's not entirely satisfactory.

An example of such sequence is the $(18,2)$-MPS sequence made of alternating $2$ and $4$.

This implies that there is exactly one $(18,2)$-MPS, which is not true, as can be seen from the next sentence:

Note that even though such a sequence is uniquely determined by $n,e$ and $a_0$.

Changing the first sentence to "... the $(18,2)$-MPS sequence $2, 4, 2, 4, \dots$", for example, would be better.

But even with such a change the notation looks very strange to me. The initial definition of $(n,e)$-MPS is not complete because $a_0$ is left out. It's only mentioned later as an afterthough in a "Note that...". Reformulating the definition as $(n,e,a_0)$-MPS would make it absolutely clear that there is a one-to-one correspondence between the triples $(n,e,a_0)$ and mirror power sequences. I don't know what purpose is served by making $a_0$ implicit and talking about $(n,e)$ pairs.

[Animus]

2, 4, 2, 4, ... is indeed a different $(18,2)-MPS$ sequence than
4, 2, 4, 2, ...

This follows from the problem statement

Note that even though such a sequence is uniquely determined by n, e and a₀, for most values such a sequence does not exist.

and is confirmed by the examples given.

The example above also indicates that $n$ and $e$ are much more coining the sequence than the start element $a_0$ (as the two sequences above differ only by an additional first element); this is the reason for naming the MPS-sequences the way we did.

[Mike]

I have wasted several days looking for the elusive 2 sequences for n <= 100. My code deliberately included a device to rotate a sequence to its lowest form in order to avoid double-counting!

When I finally gave up and peeked into this forum: Hey Presto! - all the sequences had already been found.

I fully agree that the definition offered is misleading.

[traxex]

It feels very silly to complain about a problem (repeatedly!) after solving it, but I still think the problem statement could be made much clearer with just a small change.

"An example of such sequence is the $(18,2)$-MPS-sequence made of alternating $2$ and $4$."

This sentence, in isolation, seems to say that the two sequences with $a_0=2$ and $a_0=4$ are the same. The rest of the problem statement does explain that this is not the case, but I've learned to assume that every sentence is carefully written and significant. So the contradiction is very confusing. Without the sample values I would have solved the wrong problem at first.

Changing this sentence to say "Two examples ..., with either $a_0=2$ or $a_0=4$." would resolve the issue. Alternatively, equally good would be keeping "An example ..." but stating that $a_0=2$ in this example.

[hk]

The second sentence of the problem wording has been changed into:

Examples of such sequences are the two $(18,2)$-MPS sequences made of alternating $2$ and $4$.

[castrate]

Sorry, but I failed to understand the problem. What's the meaning of "number of $(n,e)$-MPS for some $e$"? If there is an $n$ corresponding to multiple $e$'s, should I sum over all $e$'s to get $C(n)$?
For example, if $n=18$, there are two $(18,2)$-MPS as given in the problem statement, but there is also an $(18,4)$-MPS which consists of all 2's. So should $C(18)$ counted as 2 or 3? If I take $C(18)=3$(and deal with other values similarly), I would get $D(100)=23$ instead of 21.

Edit: I have realized my mistake. Thank you all.