[seqfan] Triangles of sums

jnthn stdhr jstdhr at gmail.com
Fri Jul 23 07:56:03 CEST 2021

```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/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
```