[seqfan] A091967 and A102288
kimpire at yahoo.com
Sun Nov 27 11:48:29 CET 2016
I have a question and comment about these two sequences. Quick summary: A091967(n) is the nth term of the nth sequence in OEIS, and A102288 is Cantor's diagonal method as applied to the OEIS and is therefore A091967(n)+1. A102288 is therefore a sequence that theoretically cannot appear in OEIS.
When this morning I thought of the idea behind A102288 and searched for it, I couldn't find it. I couldn't fathom how this was possible, and did dozens of searches to try and find it. Eventually I gave up, and I created an account to submit it. I reserved an A-number first (so I could reference the sequence's own A-number and mention a(A) as undefinable), but while I was filling in the form I discovered the superseeker. This located the sequence for me, and the reason I hadn't found it yet turned out to be very simple: I had the first entry wrong.
Or did I? A091967 writes "This version ignores the offset of A_n and just counts from the beginning of the terms shown in the OEIS entry", and A102288(n) is A091967(n)+1. A000001 starts with a(0)=0, yet A091967(1)=1 and A102288(1)=2. All other entries in A091967 and A102288 seem to treat a(0), if it exists, as term #1 because we're ignoring the offset. So why is it different only for a(1)?
Is this an example of M. F. Hasler's comment on A091967 that "The sequence may also change each time an additional initial term is prefixed to some other sequence, which happens quite frequently in the OEIS", and merely signifies that A000001 has been changed? Or am I doing something wrong here? I seem to have all of the other entries in the sequence correct. Normally I wouldn't hesitate to just correct it myself, but this is the *very first entry* in the encyclopedia, so the probability that it hadn't been noticed before now (and the probability that somebody wouldn't notice that the very first term in these sequences is wrong) are low enough to make me doubt myself.
Incidentally, I added a(43) and a(44) to A091967 and A102288. a(43) because A000043(43) has been discovered since somebody last updated it, and a(44) because NJAS wrote explicitly what it would be. I could have extended them further, but A000045 starts with a 0 offset, and I was nervous about making that addition until I knew the answer to the above question.
Finally, I'd like to make a proposal to slightly add to the definition of A102288. Currently, the definition does not actually bring about the quality that the diagonal method is supposed to be used to create, namely: "a sequence that cannot appear in OEIS". We have some unknown entries beginning at a(47), but even if it and all subsequent unknown entries are discovered, the sequence ends with a(52); a(53) does not exist because A000053 is finite and too short. Thus, the sequence in its entirety may in fact one day appear in its entirety in OEIS, which defeats the very purpose of using the diagonal method.
But what if we add "if A_n(n) does not exist, a(n)=1" to the definition? This lets the sequence retain its quality of being "a sequence that cannot appear in the OEIS" - because it is still not identical to any other sequence - and also enables us to theoretically extend the sequence up to the undefinable a(102288) itself. As stated, we currently reach an not-yet-known value at a(47), so this change to the definition will not yet have any practical effect, but since the sequence is not of true mathematical significance and is of the "just for fun" type anyway, it would be nice to be able to state unequivocally that the sequence cannot appear in this database.
Sadly, we obviously cannot add something similar to the definition of A(091967).
Alternatively: I can use the A-number I've already reserved to duplicate A102288 with this slightly different definition. I think that's overkill, though, since we already have *two* types of this sequence, one respecting and one ignoring the offset. We probably don't need a third.
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