# [seqfan] Re: Triangles of sums

jnthn stdhr jstdhr at gmail.com
Fri Jul 23 16:37:54 CEST 2021

```Maybe this will help...

For each element in a triangle, consider all partitions of that element
that have exactly two parts (possible children).  For n=2, place 2 at the
apex to form a one-row triangle:

2

2
1 1

This is not valid and we have no more partitions to try, so a(2) = 1

But if all children can have distinct children then we have a new and
complete row and the triangle(s) of lesser height are "incomplete."

Let's look at possible triangles for a(9), which include the first
*successful* use of backtracking:

9          9                     9
9      9      9        6 3       6 3       9        5 4
9    8 1   7 2   6 3    5 1 2    4 2 1    5 4     2 3 1

Because we *can* have more than one row the first triangle is dropped.
The next two are valid and have no possible children, so we count them.
The fourth one is dropped because the next two utilize backtracking to find
solutions with *more* than two rows.  The seventh one is dropped because we
find one solution with more than two rows.  So we end up with a(9) = 5:

9          9          9
9      9      6 3       6 3       5 4
8 1   7 2   5 1 2    4 2 1    2 3 1

Hope this helps.

-jnthn

On Fri, Jul 23, 2021, 6:26 AM jnthn stdhr <jstdhr at gmail.com> wrote:

>
>
> On Fri, Jul 23, 2021, 6:13 AM Allan Wechsler <acwacw at gmail.com> wrote:
>
>> 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, hence 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/
>> > >
>> >
>> > --
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
```