[seqfan] Re: Toothpick sequences (again) (from Richard Mathar by way of admin)

[Benoît Jubin]

2009/6/4 Benoît Jubin <benoit.jubin at gmail.com>:
>
> It might be worth looking at other dimensions:
> -2D: (A139250,A139251)
> -3D: (A160160,A160161)
> I think the 1D equivalent is (A006046,B), where B(n)=2^A000120(n).

The sequence B is actually in the OEIS: it's the Gould sequence
A001316.  The cellular automaton interpretation is mentioned there
(comment by Ben Branman).  (the entry contains a typo: in the second
line of the comments, n=2^n-1 should read n=2^k-1).

I computed the higher dimensional versions (in the sense discussed in
this thread) for dim=2,3,4,5. They don't seem to be in the OEIS.  Here
are there logarithm in base 2:
0,1,2,2,2,3,4,3,2,3,4,4,4,5,6,4,2,3,4,4,4,5,6,5,4,5,6,6,6,7,8,5,2,3,4
0,1,2,3,3,3,3,4,5,6,5,4,3,4,5,6,6,6,6,7,8,9,7,5,3,4,5,6,6,6,6,7,8,9,8
0,1,2,3,4,4,4,4,4,5,6,7,8,7,6,5,4,5,6,7,8,8,8,8,8,9,10,11,12,10,8,6,4,5,6
0,1,2,3,4,5,5,5,5,5,5,6,7,8,9,10,9,8,7,6,5,6,7,8,9,10,10,10,10,10,10,11,12,13,14

This confirms what I wrote in my previous email:

> I also think that the d-dimensional equivalent for the
> first-difference sequence begins with
> 1,2,4...2^d (no-obstruction phase), then 2^d d more times, and then
> consists only of multiples of 2^d.

Also, a graph representation is rather meaningful.  Do you have any
comments before I submit them?

Thanks,
Benoit







>
> On Wed, Jun 3, 2009 at 10:49 PM,  <franktaw at netscape.net> wrote:
>> I haven't checked in detail, but I suspect what is happening is a
>> rotation of axes.  The first toothpick is oriented with the x axis.
>> Then ones are placed on its ends, aligned with the y axis.  The third
>> generation is aligned with the z axis, and then it loops back to the x
>> axis.  As before, a toothpick is only placed when there is a single end
>> at a point, not when there are two meeting there (more than two is
>> impossible).
>>
>> Franklin T. Adams-Watters
>>
>> -----Original Message-----
>> From: Benoît Jubin <benoit.jubin at gmail.com>
>>
>> On Tue, Jun 2, 2009 at 8:03 AM, Olivier Gerard
>> <olivier.gerard at gmail.com> wrote:
>>> From Richard Mathar:
>>> Benoit wrote in
>> http://list.seqfan.eu/pipermail/seqfan/2009-May/001532.html
>>> bj> Benoît Jubin benoit.jubin at gmail.com
>>> bj> Thu May 28 02:34:32 CEST 2009
>>> bj> ...
>>> bj> Does anyone know the definitions used for the "3d toothpick
>> sequences"
>>> bj> A160160, A160161, A160120, A160121, A160170, A160171?
>>> bj>
>>> bj> The author is in copy to this email, and I'd like to politely
>> remind
>>> bj> him that submissions are not "guess-my-sequence" puzzles.
>>> bj> ...
>>> ...
>> ...
>>> Basically A160160, A160170, A160408 remain completely obscure also to
>> me.
>>
>> For me too, as well as A160120.  Anyone?
>>
>>
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>