[seqfan] Re: confused about toothpick sequence A139250!

Benoît Jubin benoit.jubin at gmail.com
Wed Apr 15 22:49:24 CEST 2009

To me, the definition is:

a(0)=0, and a(n) is the number of toothpicks at the n^th step, where
at each step you add toothpicks of length two with integer-coordinate
endpoints, and:
- at the first step you place one toothpick (anywhere),
- at the n^th step (n>1), you add to the existing configuration as
many toothpicks as possible, not overlapping existing toothpicks, and
with middle-point an endpoint of the existing configuration (not any
endpoint of any toothpick).

It is a property that (if you begin with an horizontal toothpick) at
an even step you add only vertical toothpicks and at an odd step you
add only horizontal toothpicks.  It is also a property that the
endpoints of any configuration thus obtain won't be at distance one of
each other (thus one can add the toothpicks at a given step either
simultaneously or successively, the resulting configuration will be
the same).  Actually, the endpoints of any configuration have their
sum of coordinates of the same parity as the step (if the first
toothpick is centered at a point whose coordinate-sum is even), thus
lie at a distance of each other an even integer (in the infinity

I can send a text file as an illustration, if you want.


On Wed, Apr 15, 2009 at 1:05 PM, N. J. A. Sloane <njas at research.att.com> wrote:
> There were several responses to my request for a precise definition
> of A139520, but none of them really satisfied me.
> I still do not understand the definition.
> A toothpick is a copy of the closed interval [0,1].
> Given a configuration of toothpicks in the plane.
> At the next step we add as many toothpicks as possible,
> subject to certain conditions.
> - Each new toothpick must lie in the X direction or the Y direction
> - Two toothpicks may never cross
> - Each new toothpick must have at least one of its endpoints
>    touching the midpoint of an existing toothpick
> That is not enough (it gives 1,3,7,13,21,...), so there
> must be an additional rule.  I can think of several versions,
> which is why I asked the question in the first place.
> I repeat, what is the precise definition of A139250?
> Neil
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