[seqfan] Re: confused about toothpick sequence A139250!
Benoît Jubin
benoit.jubin at gmail.com
Fri Apr 17 19:50:08 CEST 2009
On Thu, Apr 16, 2009 at 12:26 PM, <franktaw at netscape.net> wrote:
> Actually, the configurations aren't all that different. Start with
> one, rotate each toothpick a quarter-turn, and then rotate the whole
> configuration a quarter turn, and you get the other one.
Nice! So in the cellular space interpretation you described, this
would correspond to this: on an infinite chessboard, begin with one
white cell "on" (step 1), then:
- at the step 2n, turn on those black cells which have exactly one
horizontal neighbour,
- at the step 2n+1, turn on those white cells which have exactly one
vertical neighbour.
Then a(n) is the number of cells "on" after the n^th step.
Benoit
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Benoît Jubin <benoit.jubin at gmail.com>
>
> 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.
>
> Benoit
>
>
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