[seqfan] Improving Approved A() Entries
P. Michael Hutchins
pmh232 at gmail.com
Sun Mar 11 20:38:20 CET 2018
I asked an OEIS contact this:
I want to study A163778, since I discovered that it's related to my 294673,
> but I can't understand (the English of) the COMMENTS..
>
> ( Archimedes' spiral with polar equation r=c(pi + a) (c>0; a is the angle)
> defines a family of permutations: 1 is placed at the origin of the XY-plane
> and each time, as a increases, that r hits the X-axis we put the next
> number on the X-axis; reading the numbers from left to right yields the
> permutation of 1..N (N>=2) given by N-1 N-3 ... 5 3 1 2 4 6 ... N-1 N (if N
> is even) or ...)
>
> I don't think it's just me.
> Although I'd love to just be told what it means, I think we ought to fix
> it.
> I didn't find the author's email address.
...and I got this (good) response:
>
>
> The full comment is:
> Archimedes' spiral with polar equation r=c(pi + a) (c>0; a is the angle)
> defines a family of permutations: 1 is placed at the origin of the XY-plane
> and each time, as a increases, that r hits the X-axis we put the next
> number on the X-axis; reading the numbers from left to right yields the
> permutation of 1..N (N>=2) given by N-1 N-3 ... 5 3 1 2 4 6 ... N-1 N (if N
> is even) or N N-2 ... 5 3 1 2 4 6 ... N-3 N-1 (if N is odd). Then N is an
> A_1-prime if this permutation consists of a single cycle of length N. So
> all A_1-primes are odd.
>
> I agree that the meaning of the comment is unclear but I am sure it can be
> unravelled. Looking at https://en.wikipedia.org/wiki/Archimedean_spiral,
> I think that the most relevant fact is that successive turns of the spiral
> are equidistant from one another, in particular on the x-axis. My best
> guess is that each time the spiral crosses the X-axis the next integer in
> the sequence {1..N} not already allocated is allocated to the crossing
> point to construct a permutation of {1..N} when read from left to right.
> However, I do not see how a ”family” of permutations is generated. The
> example permutation appearing in the comment contains a mistake for the
> case when N is even; the second (N-1) should be (N-2). The example is only
> relevant for the spiral that is the mirror image of the one in the first
> diagram of https://en.wikipedia.org/wiki/Archimedean_spiral. I don’t see
> how the conclusion “Then N is an A_1-prime if this permutation consists
> of a single cycle of length N. So all A_1-primes are odd.” follows at all!
>
...so my overall question is What procedures are in place to fix uch things.
(Don't we want to fix them?)
Thanks,
-- M.
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