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

Tue Sep 9 17:13:09 CEST 2014

Seems reasonable to me.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Tue, Sep 9, 2014 at 5:30 AM, Veikko Pohjola <veikko at nordem.fi> wrote:

> 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
> >>>>>>
> >>>>>> _______________________________________________
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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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