[seqfan] A simple yet mysterious sequence

[Antti Karttunen]

On 3/28/22, Neil Sloane <njasloane at gmail.com> wrote:
> 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).

Thanks for the reference. I just found the book to be freely available in
https://archive.org/details/in.ernet.dli.2015.134264/mode/2up
in various formats.

Also, it has good reviews in Amazon:
https://www.amazon.com/Hadamard-transform-optics-Martin-Harwit-ebook/dp/B01D4OA6DY?


Best,

Antti

>
> 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/
>> >
>>
>