[seqfan] Re: Triangles of sums

[Nacin, David]

Hi Jonathan,

I'm not sure I understand, since I seem to be getting different numbers than you.  For a(9), I get either 15 or 8 depending on whether you're counting up to y-axis symmetry .  I made a pic to explain.

Also should your

  8
 6 4
5 1 3

from the e-mail actually be

  10
 6 4
5 1 3

instead?  I'm getting no solutions for 8 with that height.

Two more quick questions about symmetry and depth.  Are we counting up to reflection?  I'm assuming so since you have a(3) = 1 and both

 3
1 2

and

 3
2 1

are options.

Are there any requirements for height?  For a number like 9 at the peak, do we count the height 1, height 2 and height 3 all together?

-David

________________________________
From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of jnthn stdhr <jstdhr at gmail.com>
Sent: Wednesday, July 21, 2021 12:06 PM
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Triangles of sums

Hello seqfans.

Long time no sequence (apologies.)  Inspired by , https://nam11.safelinks.protection.outlook.com/?url=http%3A%2F%2Foeis.org%2FA340389&data=04%7C01%7Cnacind%40wpunj.edu%7Ce6cdb18bb7e547c4ed8d08d94d5489b6%7C74540637643546cc87a46d38efb78538%7C0%7C0%7C637625847764776419%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=G8m6%2F0h0f31cSsYeWXFAraTmPvN%2Bt%2Ft7Bf%2Flzd%2FaOLc%3D&reserved=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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