[seqfan] Re: Triangles of sums
Nacin, David
NACIND at wpunj.edu
Fri Jul 23 00:18:14 CEST 2021
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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