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

[Alex Meiburg]

It should be mentioned at least once in this discussion that the
observation Pohjola made, that iterating the function always leads to 0 or
1, is far from certain, and would be fascinating if true. Although I
verified it for numbers from -10^7 to 10^7, to prove it always true would
require deep statements about the irrationality of pi/2 -- and the
non-monotonicity of its behavior is reminiscent of the headache brought on
by the Collatz conjecture. The statement that it always reaches one of
these two, I would wager, would probably have difficulty on par with such
conjectures, or interesting open problems like the convergence of
http://mathworld.wolfram.com/FlintHillsSeries.html .

>From a "probabilistic" viewpoint it makes sense; a number n has O(1/n^2)
probability of being the output of Floor[Tan[x]] for random chosen x, so
the chance at any number, say, over 1000 produces itself as the output, is
the sum 1/ 1000^2 + 1/1001^2 + 1/1002^2 ... which converges to a finite
probability. This gives the notion that "statistically" there's a good
chance of no fixed points. (With nothing to say about possible 2-loops or
3-loops.) Indeed, if you looked at the behavior of Floor[Tan[pi*a*n]] for
some irrational a, choosing a randomly would lead to this probability of
encountering fixed points. This doesn't tell us much about what happens in
the case of a = 1/pi, of course. :P


-- Alexander Meiburg


2014-09-04 16:40 GMT-07:00 Frank Adams-Watters <franktaw at netscape.net>:

> 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
>>>>>
>>>>> _______________________________________________
>>>>>
>>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>>
>>>>
>>>>
>>>>
>>>> --
>>>> 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
>>>>
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>>>>
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