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

[Veikko Pohjola]

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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