[seqfan] Re: Triangles of sums

[Allan Wechsler]

Okay, that list helps me focus on my area of confusion. Why do you exclude
a singleton 3? You allow a singleton 2. What is the rule?

On Fri, Jul 23, 2021, 5:40 AM jnthn stdhr <jstdhr at gmail.com> wrote:

> Hi, Allan.
>
> To clarify, 1 and 2 have no possible distinct children, hense they have
> height of 1.  And you are correct, I am not counting reflections, since
> A340389 does not.
>
> The first few triangles are:
>
>             3      4      5      5
> 1,  2,  1 2,  1 3,  1 4,  2 3
>
> As for the typo, In my notebook see I have the 10, 64, 513 triangle just
> below the 8, 53, 412 triangle, so I think my error is a result of looking
> at the wrong line and not seeing the obvious error 6+4!=8. Sorry for the
> confusion.
>
> -jnthn
>
>
> On Thursday, July 22, 2021, Allan Wechsler <acwacw at gmail.com> wrote:
>
> > I applaud your instinct to make sure that simple cases accompany their
> more
> > complicated brethren into the Encyclopedia -- I think this is right on
> > target.
> >
> > But I am missing something here. Can you display all the triangles for
> n=1
> > to 3? My problem is that you must be allowing triangles of one row in
> order
> > to have one example for n=1 and n=2, but then it seems to me that you
> ought
> > to have two examples for n=3, one with one row, and one with two rows.
> But
> > you say there is only one.
> >
> > Also, I am assuming you do *not* consider reflections around the vertical
> > axis to be distinct solutions.
> >
> > I'm sure some sequence fanatic will be happy to help you as soon as it's
> > clearer what your definitions are.
> >
> > One last thing: 6 + 4 does not equal 8, as your second example seems to
> > claim.
> >
> > On Thu, Jul 22, 2021 at 5:06 PM jnthn stdhr <jstdhr at gmail.com> wrote:
> >
> > > Hello seqfans.
> > >
> > > Long time no sequence (apologies.)  Inspired by ,
> > http://oeis.org/A340389
> > > wondered if a generalized sequence, the number of sum triangles of n,
> > was
> > > in the database -- it appears it is not.
> > >
> > > If we define a sum triangle of n as a triangle with n at its apex, all
> > > pair-wise members (x, y) of rows 2,3,4,... sum to the element
> immediately
> > > above, every element is distinct, and rows are complete (length of row
> m
> > =
> > > length of row (m-1) + 1.
> > >
> > > For example:
> > >
> > >           8         9        9
> > >  3      6 4      6 3     6 3
> > > 2 1   5 1 3   5 1 2  4 2 1
> > >
> > >
> > > The sequence I get for n=1 to 30 is:
> > >
> > >
> > > [1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 9, 11, 11, 18, 17, 22, 23, 29, 31,
> 38,
> > > 37, 46, 49, 58, 59, 72, 76, 86, 90]
> > >
> > > My python code is about 70 lines long.  Maybe a MMA expert could write
> a
> > > more concise program and confirm the the sequence?
> > >
> > > -Jonathan
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
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> >
>
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