[seqfan] Re: Triangles of sums
Allan Wechsler
acwacw at gmail.com
Fri Jul 23 17:43:21 CEST 2021
Now I understand. You only count "maximal" triangles, ones that cannot be
extended by another row. (3) doesn't count because (3; 1 2) exists, and (9;
6 3) doesn't count because (9; 6 3; 5 1 2) exists.
Does the sequence counting *all* such triangles already exist? If it does,
it should be linked.
I might write a program to see if I can confirm your numbers, now that I
understand your rules.
On Fri, Jul 23, 2021 at 10:44 AM jnthn stdhr <jstdhr at gmail.com> wrote:
> 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
>
> Now try adding new row(s):
>
> 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/
> >> > >
> >> >
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> >> >
> >>
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> >>
> >
>
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