[seqfan] Re: Number of orders of distances to vertices of n-dimensional cube

Lucas Brown lucasbrown.cit at gmail.com
Tue Nov 15 08:54:05 CET 2022


Offline execution of that code yielded a(5) = 1981440 in about 20 minutes.

On Mon, Nov 14, 2022 at 7:16 PM Max Alekseyev <maxale at gmail.com> wrote:

> Hi Pierre,
>
> The next term is 5376 as computed by this quick'n'dirty Sage code based on
> hyperplane arrangements:
> https://sagecell.sagemath.org/?q=gbndab
>
> It may also be possible to compute a(5) but not online.
>
> Regards,
> Max
>
>
> On Mon, Nov 14, 2022 at 9:08 AM Pierre Abbat <phma at bezitopo.org> wrote:
>
> > Let C be an n-dimensional cube and p be a point in R^n such that the
> > distances
> > from p to the 2^n vertices of C are all different. Write the vertices in
> > order
> > of their distance from p. How many different orders of vertices are
> there?
> >
> > I'm writing an algorithm to search an octree for the two closest points
> to
> > a
> > given point p. Usually one of them will be p. I search the eight subcubes
> > in
> > order of their distance from p, so that I can ignore farther subcubes
> > entirely, knowing that the distance to the closest points is less than
> the
> > distance to the subcube. There are 96 orders, but I'm using only 48 to
> > simplify the code, using e.g. {0,1,2,3,4,5,6,7} but not
> {0,1,2,4,3,5,6,7}.
> >
> > The sequence begins 1,2,8,96. There are four sequences beginning
> 1,2,8,96,
> > of
> > which one has offset 1,2 so I discard it. I suspect it's the map-folding
> > sequence A001417, since the next term 4608 is divisible by 2^4×4!, but my
> > four-dimensional intuition is not good enough to verify it without
> writing
> > some code, and I need to write only the code for n=3 because someone
> needs
> > it
> > for a device.
> >
> > Take a set of n real numbers {x[0]..x[n-1]}. Take all 2^n subsets of the
> > set,
> > sum each subset, and make sure they're all different. The nth term of the
> > sequence (starting at n=0) is the number of different orders the sums can
> > be
> > in.
> >
> > Pierre
> > --
> > Don't buy a French car in Holland. It may be a citroen.
> >
> >
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>



More information about the SeqFan mailing list