[seqfan] Re: Triangles of sums

jnthn stdhr jstdhr at gmail.com
Fri Jul 23 21:51:01 CEST 2021


I can add the "count everything version".  As for record heights, I think
no matter what you are counting the sequence of record heights will be the
same, no? And, as I mentioned earlier, that sequence exists. The final term
in that sequence is ten digits long, which is why I wish a program had been
included.

On Friday, July 23, 2021, Nacin, David <NACIND at wpunj.edu> wrote:

> This was helpful!  You're counting all the non-extendable pyramids -
> pyramids for which it is not possible to add a row below? You're also
> counting up to reflective symmetry, counting a pyramid and its reflection
> only once?  You get a(8) = 3 and a(9) = 5 because you are only counting the
> blue triangles in the two attached pictures since those are the
> non-extendable ones up to symmetry.
>
> The way I see it then, there are six sequences of number of pyramids here.
>
> There are the non-extendable ones which you've been counting.  There are
> the max-height pyramids which only count ones that achieve the maximum
> height for that n.  There are the any-height pyramids which include
> pyramids of all heights like I was counting in my previous e-mail.  Then
> for each of those three you also have the two cases of counting up to
> reflections or counting all the pyramids.
>
> I have code that goes pretty far for four of these, tackling the two
> max-height and two any-height cases.  I could quickly modify it to do the
> other two cases as well.  My code can go out to the first hundred terms or
> so, but then gets slow, though I could probably speed it up given some
> time.  Also, I'm sure someone else could write something faster if they
> desired.
>
> -David
>
> -----Original Message-----
> From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of jnthn stdhr
> Sent: Friday, July 23, 2021 10:38 AM
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Triangles of sums
>
> 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 ,
> >> > > https://nam11.safelinks.protection.outlook.com/?url=http%3A%2F%2F
> >> > > oeis.org%2FA340389&data=04%7C01%7Cnacind%40wpunj.edu%7Ccd7414
> >> > > 6129354ac9e98208d94de86038%7C74540637643546cc87a46d38efb78538%7C0
> >> > > %7C0%7C637626482721186717%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjA
> >> > > wMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sd
> >> > > ata=ePIZykhmOcd90B%2F7rnjR7ekcPjf43DWjCvI%2FuIKjZ7c%3D&reserv
> >> > > ed=0
> >> > > > 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
> >> > > >
> >> > > > --
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> >> > > > 8%7C0%7C0%7C637626482721186717%7CUnknown%7CTWFpbGZsb3d8eyJWIjoi
> >> > > > MC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C10
> >> > > > 00&sdata=TwwVPb0xDEy2RL9bVeAMpypyVh7KyR2kBwnZX1oHp%2FY%3D&a
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> >> > > 7C0%7C637626482721196672%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAw
> >> > > MDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sda
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> >> > IjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=89RcRJ2
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