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

Benoît Jubin benoit.jubin at gmail.com
Thu Jun 4 06:03:44 CEST 2009


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> ..
>
>
> One might define a 3D version of the toothpick sequence by
> (i) place the first toothpick aligned with any of the Cartesian axes of the
>  cubic grid.
> (ii) For the step from generation n to n+1 define the exposed
>  toothpick ends as those 3D points which are one of the two end points
>  defined by one (and only one, not shared by another) of the toothpicks in
>  generation n and which are not a midpoint of any toothpick of generation n.
> (iii) Add in generation n+1 at each of the exposed points marked
>  in step (ii) a cross (=of two toothpicks) in the plane perpendicular to the
>  toothpick that defined the exposed point.
>
> This should (I guess, should be checked !) start the count of
> toothpicks (crosses
> contain 2 of them, counted individually) as
>
> 0,1,5,21,37,53,69,85,101,117,133,149,165,181,197,213,229,245,261,277,293,309,
> 325,341,357,373,389,405,421,437,453,469,485,501,517,533,549,565,581,597,613,
> 629,645,661,677,693,709,725,741,757,773,789,805...

I didn't check the sequence, but for n>1, this is 16*(n-2)+5, which
would be a less interesting behaviour than in 2D.  What about higher
dimensions?

> Basically A160160, A160170, A160408 remain completely obscure also to me.

For me too, as well as A160120.  Anyone?

Benoit




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