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

[Charles Greathouse]

I believe the comment (and its typo) are mine. The result falls out
directly from the equidistribution of the natural numbers mod Pi. I believe
this is known as the equidistribution theorem (a special case of the
ergodic theorem).

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Thu, Sep 4, 2014 at 3:37 PM, Veikko Pohjola <veikko at nordem.fi> wrote:

> 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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