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

Tue Sep 9 11:30:17 CEST 2014

Would it be appropriate to conjecture that applying the function Floor[Tan] k times, with k sufficiently large, on the finite sequence Floor[Tan[n]], n=0...N, the result is a sequence composed of 0’s and 1’s for all values of N.
Veikko

Alex Meiburg kirjoitti 5.9.2014 kello 5.32:

> 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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