[seqfan] Re: confused about toothpick sequence A139250!

Benoît Jubin benoit.jubin at gmail.com
Thu Apr 16 11:44:20 CEST 2009

Actually, two "dual" definitions give this sequence: the one I gave
below (which is essentially the same as Rob Pratt's), and Neil's, when
you add toothpicks such that exactly (and not "at least") one of their
endpoints is the middle of an existing toothpick.  The two sequences
of configurations obtained are different, but the numbers of
toothpicks are the same (at least up to a(8)=43).  This is a
noteworthy fact, and I don't see an immediate argument to prove it.

The sequence corresponding to Neil's original definition (with "at
least") is n^2-n+1, because there will be no hole left in the grid.


2009/4/15 Benoît Jubin <benoit.jubin at gmail.com>:
> 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
> norm).
> I can send a text file as an illustration, if you want.
> Benoit
> 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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