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

[Veikko Pohjola]

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
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