[seqfan] tan(n). Was: Into subtleties of musical information

Tue May 26 13:21:57 CEST 2015

On Tue, May 26, 2015 at 12:39 PM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> Hmm, just realized that there shouldn't be any a priori reason that
> floor(tan(n)) could not have also other fixed points than just 0 and
> 1.
> For example, we have A000503(37362253) = 37754921, and 37754921 -
> 37362253 = 392668 (just one percent difference between the argument
> and the result).
>
> So, I have to define some of these sequences more defensively,
> allowing the possibility for other fixed points than 0 or 1 as well.

I think there are three alternative options how to proceed:

1) Leave the definitions of

https://oeis.org/draft/A258022
"Natural numbers n with property that starting from them, the map x ->
floor(tan(x)) reaches 0 (instead of 1) as the eventual fixed point. "
and
https://oeis.org/draft/A258024 (same but with fixed point value 1 instead 0)

as they are, and then cross-reference these as "conjectured
complements" of each other (like I have done now)

OR

2):

Leave the definition of A258022 as it is,
but change A258024 to include all numbers which eventually reach a
nonzero fixed point.

OR

3):

Change the both, A258022 all numbers which reach a fixed point <= 0,
and A258024 for all numbers which reach a fixed point > 0.

Which one to choose? Most probably this does not have any effect on
any actual terms computed, but would just make the mathematical
definitions more consistent.

I also started searching cases where A000503(n) > n, and also the
cases where the ratio A000503(n)/n obtains values successively nearer
to one.
I got this far:

A000503(4) = 1, ratio = 4.
A000503(14) = 7, ratio = 2.
A000503(344) = 227, ratio = 1.5154185022026432
A000503(51819) = 40589, ratio = 1.2766759466850623
!!! A000503(260515) = 383610, 383610 >= 260515
A000503(4846147) = 3910993, ratio = 1.239109095822979
A000503(37362253) = 37754921, ratio = .9895995544527825

then realized/recalled that this topic was discussed some months ago:

https://oeis.org/A088306 "Integers n with tan n > |n|, ordered by |n|. "
1, -2, -11, -33, -52174, 260515, -573204, 37362253, -42781604, 122925461, ...

https://oeis.org/A249836 "Numbers for which tan(n) > n."
1, 260515, 37362253, 122925461, 534483448, 3083975227, ...

Is this last one a subsequence of

https://oeis.org/A096456 "Numerators of convergents to Pi/2."
1, 2, 3, 11, 344, 355, 51819, 52174, 260515, 573204, 4846147, 5419351,
37362253, ... ???
(would make some sense...)

Best regards,

Antti

>
> Best,
>
> Antti
>
>
>
>
> On Tue, May 26, 2015 at 8:47 AM, Veikko Pohjola <veikko at nordem.fi> wrote:
>> It is worth mentioning that the following numbers seem to appear in the sequence of differences:
>> 3, 6, 7, 9, 10, 12, 13, 16, 19.
>> Each of them has its own density and distribution exhibiting its own rhythmic pattern and thus resulting in polyrythmic music, a phenomenon studied by Robert Walker, for instance in the case of Fibonacci numbers (See reference in A000045).
>> Veikko
>>
>> Antti Karttunen kirjoitti 25.5.2015 kello 5.42:
>>
>>> Veikko wrote in http://list.seqfan.eu/pipermail/seqfan/2015-May/014904.html
>>>
>>>> Well, I did not refer directly to the values given in A258024, which are the n’s which reduce to 1 instead of 0, but to their differences, which of course is a separate, though related, sequence
>>>> 3, 19, 3, 19, 3, 19, 3, 19, 3, 13, 6, 3, 7, 6, 6, 3, 7, 6, 6, 3, 13, 6, 3, 3, 10, 6, 3, 3, 10, 6, 3, 3, 16, 3, 3, 16, 3….
>>>
>>> This is now in https://oeis.org/draft/A258200
>>>
>>>> In the search of the pattern this latter is more informative as can be readily seen. It may well deserve a separate sequence, especially if its musicality either as such or due to the potential of drawing conclusions on its basis has some interest. I can submit it and then we’ll see.
>>>
>>>> Then there is a possibility for a sequence of n’s which reduce to 0 instead of 1,
>>>
>>> This is now https://oeis.org/draft/A258022
>>>
>>>> and their differences…
>>>
>>> It is:
>>> 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1,
>>> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1,
>>> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1,
>>> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
>>> 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1,
>>> 1, 2, 1, 2
>>> but I didn't create that yet.
>>>
>>> Also, I created
>>> https://oeis.org/draft/A258021
>>> "Eventual fixed point of map x -> floor(tan(x)) when starting the
>>> iteration with the initial value x = n."
>>>
>>> and
>>>
>>> https://oeis.org/draft/A258020
>>> "Number of steps to reach either 0 or 1 with map x -> floor(tan(x))
>>> when starting iteration with the initial value x = n. "
>>>
>>> Now, because I don't trust that in MIT/GNU Scheme (floor->exact (tan
>>> n)) would not at some point be one off because of the loss of the
>>> precision, I didn't try to compute b-file for any of these sequences.
>>> Could somebody with real CAS (and the knowledge how tweaking various
>>> parameters affects that risk) do that, up to say a few thousands at
>>> least? (For A258020 and A258200 at least).
>>>
>>>
>>> Regards,
>>>
>>> Antti
>>>
>>>
>>>
>>>
>>>
>>> On Mon, May 25, 2015 at 12:56 AM, Antti Karttunen
>>> <antti.karttunen at gmail.com> wrote:
>>>> On Sun, May 24, 2015 at 12:00 AM,  <seqfan-request at list.seqfan.eu> wrote:
>>>>
>>>>> ------------------------------
>>>>>
>>>>> Message: 16
>>>>> Date: Sun, 24 May 2015 00:00:32 +0300
>>>>> From: Veikko Pohjola <veikko at nordem.fi>
>>>>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>>>>> Subject: [seqfan] Musical information about a sequence
>>>>> Message-ID: <2C234C13-4D18-4D6C-AC08-E57A1A0DAFB0 at nordem.fi>
>>>>> Content-Type: text/plain; charset=windows-1252
>>>>>
>>>>> Dear seqfans,
>>>>>
>>>>> Applying floor(tan(n)) repeatedly, a limiting sequence results composed of 0’s and 1’s only. The proportion of 1’s is somewhat over 12% and they distribute interestingly forming a repeating pattern. Converting the sequence of distances between the positions of 1’s into music (say piano) turns it to fascinating music manifesting a steady beat with a theme and delicate variations. Counting the number of beats in a sequence of known length permits to assess the number of individual sounds (terms) within each measure to about 42.
>>>>>
>>>>> I am wondering whether such a steady beat could be inherited from the periodic nature of the mother function (tan) and if so, should the length of the pattern thus be predicted.
>>>>> And in general, are this sort of musical findings regarded to belong to recreational domain and not at all to hard mathematics, not ...
>>>>
>>>> Dear Veikko,
>>>>
>>>> I don't care what other people think about what is "hard enough
>>>> mathematics" (some people have very restrictive biases), but think
>>>> that your find is very interesting.
>>>>
>>>> I have been myself trying to find good examples of the general idea
>>>> behind Per Nørgård's "infinity sequence"
>>>> http://oeis.org/A004718
>>>> "invented in an attempt to unify in a perfect way repetition and variation".
>>>> (See also https://oeis.org/A056239 for another Danish comment in
>>>> another, not related sequence. Also https://oeis.org/A126759 )
>>>>
>>>> In other words, anything on the sweet but rare region between (too
>>>> much) regularity (most base-sequences) and (too much) chaos. (Compare
>>>> also to some Wolfram's CA-classifications, although I'm not now
>>>> interested about Turing-capability. Also, it seems that human mind
>>>> cannot relish complete chaos until it is regularly repeated and thus
>>>> "amplified"?)
>>>>
>>>> So far, my attempts have concentrated on "entanglement-permutations"
>>>> and "beanstalk-sequences" (my neologisms but not my inventions) both
>>>> of which mix together a repeating pattern with some "new material",
>>>> although in different ways. I haven't yet much experimented of
>>>> actually producing any sounds of these, except some random playing
>>>> with "Listen-button" which leaves much to be desired regarding the
>>>> actual mapping, not just to notes but to rhythm/dynamics as well (or
>>>> maybe I should learn to use its various options better?) In any case,
>>>> maybe it's better to leave their exact mapping to rhythm and sounds to
>>>> more musical talents, and for me to just keep on producing more
>>>> patterns and hope that some of them are mathematically interesting and
>>>> useful as well.
>>>>
>>>>
>>>>> ... even when some useful mathematical information could be obtained by listening.
>>>>
>>>> For this, please see also:
>>>> http://www.moz.ac.at/sem/lehre/lib/bib/software/cm/Notes_from_the_Metalevel/chaos.html
>>>>
>>>>
>>>>
>>>> Terveisin,
>>>>
>>>> Antti
>>>>
>>>>>
>>>>> Best regards,
>>>>> Veikko Pohjola
>>>>>
>>>
>>