[seqfan] Re: Nice new seq needs a formula

[Allan Wechsler]

The two sequences both have that lines-going-through-square-lattices vibe,
and the growth rate looks like what one would expect, so I would bet a fair
amount of money on this conjecture being true at this point.

So we should think about proof approaches. What it feels like to me is that
there is some kind of duality relationship going on: lines through the nXn
lattice in A114043 correspond to points in the nX1 rectangle in A306302;
the set of lines that go through a lattice point in A114043 are dual to a
line in A306302. When the dividing line crosses over a lattice point in
A114043, the corresponding point moves into another region in A306302. Can
anybody rigorize this better? I'm sure there's a proof here.

On Thu, Feb 7, 2019 at 5:20 PM Neil Sloane <njasloane at gmail.com> wrote:

> To get the offsets to match, the conjecture is that
>
> A306302(n) =  n + (A114043(n+1)-1)/2
>
> E.g. 46 = 3 + (87-1)/2.
>
>
> On Thu, Feb 7, 2019 at 5:08 PM Neil Sloane <njasloane at gmail.com> wrote:
>
> > The new sequence that I mentioned , A306302(n),
> > appears to equal n + (A114043(n)-1)/2, and A114043 has an explicit
> formula
> > due to Max Alekseyev.  The two sequences A306302 and A114043
> > are sufficiently similar that it should not be hard to find a proof.
> >
> > (Thanks to Superseeker for the hint).
> >
> >
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, President, OEIS Foundation.
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> > Phone: 732 828 6098; home page: http://NeilSloane.com
> > Email: njasloane at gmail.com
> >
> >
> >
> > On Thu, Feb 7, 2019 at 1:26 PM Hugo Pfoertner <yae9911 at gmail.com> wrote:
> >
> >> Links to available literature for this problem are given in
> >> http://oeis.org/A288177
> >> http://oeis.org/A306302 is part of the triangle http://oeis.org/A288187
> >> If more terms can be found - fine, but we shouldn't re-invent the
> reseach
> >> done by Pfetsch, Ziegler and others 15 years ago.
> >>
> >> Hugo Pfoertner
> >>
> >> On Thu, Feb 7, 2019 at 7:16 PM Neil Sloane <njasloane at gmail.com> wrote:
> >>
> >> > PS and what about the numbers of vertices (including both boundary
> >> points
> >> > and interior intersection points)?
> >> >
> >> >
> >> > On Thu, Feb 7, 2019 at 1:09 PM Neil Sloane <njasloane at gmail.com>
> wrote:
> >> >
> >> > > No of regions in a strip of rectangles:  A306302
> >> > >
> >> > > Given enough terms, will probably be guessable
> >> > > 4, 16, 46, 104, 214 is not enough
> >> > >
> >> > >
> >> > >
> >> >
> >> > --
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> >> >
> >>
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> >>
> >
>
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