[seqfan] Re: A simple yet mysterious sequence

[Neil Sloane]

Well, funny you should ask, there's the book I wrote with Martin Harwit,
Hadamard Transform Optics !   We show how to use Hadamard transforms to
improve accuracy of measuring the spectra of light (and which,
incidentally, is being used in the James Webb Telescope, a million miles
from here).

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com



On Mon, Mar 28, 2022 at 11:05 AM Antti Karttunen <antti.karttunen at gmail.com>
wrote:

> On 3/28/22, Neil Sloane <njasloane at gmail.com> wrote:
> > Lexicographically Earliest Sequences like the EKG sequence A064413 have
> > interested me for years, but are still mysterious. Each of them has a
> > set-theoretic analog, and these are even more mysterious,
> >
> > because their graphs all look much the same but are totally unexplained.
> >
> >
> > Here's the simplest of them, A109812, defined by:
> >
> > a(1)=1; thereafter a(n) = smallest positive integer not among the earlier
> > terms of the sequence such that a(n) and a(n-1) have no common 1-bits in
> > their binary representations.
> >
> > 1, 2, 4, 3, 8, 5, 10, 16, 6, 9, 18, 12, 17, 14, 32, 7, 24, 33, 20, 11,
> 36,
> > 19, ...
> >
> >
> > Look at the graph: https://oeis.org/A109812/graph  Look at the bottom
> > graph, showing 10K terms.
> >
> > (It looks remarkably like the graphs of A252867, A338833, A352200,
> A305369,
> > etc., which have much more complicated definitions.)
> >
> > I don't even know how to describe these graphs in words. They have a
> > fractal-like structure, certainly.
> >
> >
> > So here is my question: A109812 has a really simple definition: Can
> someone
> > find a recurrence or generating function, or a Discrete Fourier Transform
> > (a representation in terms of Walsh functions, perhaps),
>
> A bona fide question: Are there any such analyses (done either with
> DFT or Walsh-Hadamard transformation) successfully applied to an OEIS
> sequence that yielded additional insights? I ask this so that I could
> find a good in-ramp to the subject, in the subject matter (OEIS
> sequences) that I might understand.
>
> Also, could you recommend a good primer (a book, preferably) for
> learning about Hadamard-Walsh transformation, its applications and so
> on?
>
>
> Best regards,
>
> Antti
>
>
> > or any other explanation for the graph?
> >
> >
> > The classic Lex. Earliest Seqs. are complicated because they involve
> > properties of the primes, so that is not a surprise.  But A109812 has
> > nothing to do with primes. What is going on?
> >
> >
> > (I'm sending this to the Sequence Fans and Math Fun lists, apologies for
> > duplicate postings.  But I really want to know the answer.)
> >
> >
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, Chairman, OEIS Foundation.
> > Also Visiting Scientist, Math. Dept., Rutgers University,
> > Email: njasloane at gmail.com
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>