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

Olivier Gerard olivier.gerard at gmail.com
Tue Jun 2 17:03:03 CEST 2009

```  From Richard Mathar:

Benoit wrote in http://list.seqfan.eu/pipermail/seqfan/2009-May/001532.html
bj> BenoÃ®t 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...

The 5 after the 1 means that we have added 2 crosses at both terminal
points of the
first lone toothpick. These two crosses have 2*4=8 new exposed points, which
call for 8 new crosses (16 toothpicks) at the next stage, which defines 5+16=21
etc.

...obviously *not* what's counted in A160170.
Basically A160160, A160170, A160408 remain completely obscure also to me.

A C++ program is available to generate the sequence. Regrettably some
software in the seqfan administration seems to dispose of submissions that
contain these, so I am not attaching the coding.

Richard

```