[seqfan] Re: Floor[Tan[n]]

[Frank Adams-Watters]

Every integer means every integer. If it meant every value that occurs 
in the sequence, it would say so.

But I don't see what your problem is. (2) implies (1); so if there is a 
proof using (1), then (2) will be sufficient for that proof, too.

Franklin T. Adams-Watters

-----Original Message-----
From: Veikko Pohjola <veikko at nordem.fi>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Thu, Sep 4, 2014 2:37 pm
Subject: [seqfan] Re: Floor[Tan[n]]


I come back to my original question. I still feel that I did not get 
the answer.
To me (and actually to some of my colleagues) the statement "Evey 
[should be
every, of course, but that is not the point] integer appears infinitely 
often"
is most vague. What is referred to by "every integer"? (1) All integers 
which
once appear in the sequence, or (2) all integers which exist? If it is 
the (1),
then it is easy to understand (at least to me) why the sequence reduces 
to the
limiting sequence composed of 0's and 1's when nesting the function 
Floor[Tan],
otherwise not.
Veikko

Veikko Pohjola kirjoitti 3.9.2014 kello 12.03:

> The piece of sequence I gave is of  course that of the differences 
between the
positions of 1's.
> The positions themselves go like this
> 2, 5, 24, 27, 46, 49, 68, 71, 90, 93, 106, 112, 115, 122, 128, 134, 
137, …
> (Not in OEIS)
> Veikko
>
>
>
> Veikko Pohjola kirjoitti 3.9.2014 kello 11.50:
>
>> I decided to pose these questions having played with another 
sequence, which
forms when nesting Floor[Tan] to A000503 sufficiently many times. The 
end result
is a sequence composed of 0’s and 1’s. The position on 1’s in this 
sequence does
not seem to follow any regular pattern.
>>
>> After applying the Floor[Tan] 8 times to Floor[Tan[n]], n=0...10^5, 
the
position of 1’s in the limiting sequence (not changing when applying 
the
function the 9th time) is as follows:
>> 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, 3, 19, 3, 7, 12, 3, 19, 3, 
7, 12, 3,
19, ...
>> Interesting, huh?
>> Veikko
>>
>>
>> Neil Sloane kirjoitti 3.9.2014 kello 10.18:
>>
>>> well, tan(n) = tan(n +2Pi), right?
>>>
>>> and as n varies, n mod 2Pi will be dense in 0 to Pi
>>>
>>> On Wed, Sep 3, 2014 at 3:00 AM, Veikko Pohjola <veikko at nordem.fi> 
wrote:
>>>> Dear seqfans,
>>>>
>>>> It is obvious that Tan[x] covers all real numbers (I guess). It 
may be
obvious that Floor[Tan[x]] covers all natural numbers. But it is pretty 
far from
obvious, to me, that even Floor[Tan[n]], where n is an integer from 
zero to
infinity, would also cover all natural numbers.
>>>>
>>>> Is it this last statement above, what is meant by the comment 
”Evey integer
appears infinitely often.”, which appears in A000503? If it is, could 
and
shouldn’t it be provided with a justification in the case of being a 
conjecture,
or with a proof in he case of being a postulate?
>>>>
>>>> Veikko
>>>>
>>>> _______________________________________________
>>>>
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>>>
>>>
>>>
>>> --
>>> Dear Friends, I have now retired from AT&T. New coordinates:
>>>
>>> Neil J. A. Sloane, President, OEIS Foundation
>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>> Also Visiting Scientist, Math. Dept., Rutgers University, 
Piscataway, NJ.
>>> Phone: 732 828 6098; home page: http://NeilSloane.com
>>> Email: njasloane at gmail.com
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
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