[seqfan] Re: Triangles of sums

Nacin, David NACIND at wpunj.edu
Fri Jul 23 02:49:13 CEST 2021


Johnathan,

Actually, the first pyramid of height three that I found occurs at n=8.

  8
 3 5
2 1 4

but this brings up the question of how far out you need to go to reach a given height.  A related interesting sequence might be the smallest value a(n) for which we can generate a pyramid of height n.

For this sequence I get a(1) = 1, a(2) = 3, a(3) = 8, a(4) = 20, a(5) = 43, a(6) = 98 and so on, by the way that I'm calculating them, though I realize I might be calculating things differently since I'm getting 1, 1, 3, 3, 5, 5, 7, 9, 15, 15, 21, 25, 31, 31, 47, 45, 57, 59, 73, 83, 95, 99, 121, 133, 153, 171, 197, 219, 243, 263, 305, 331, 375, 397, 451, 499, 535, 573, 659, 701, 757, 805, 899, 957, 1071, 1089, 1219, 1293, 1381, 1471, 1641, 1699, 1835, 1923, 2121, 2179, 2413, 2475, 2705, 2885, 3067, 3147, 3485, 3591, 3895, 4009, 4367, 4553, 4903 for the total number of triangles (not counting up to y-axis symmetry.)  Adding 1 and dividing by 2 gives the resulting sequence up to symmetry giving us 1, 1, 2, 2, 3, 3, 4, 5, 8, 8, 11, 13, 16, 16, 24, 23, 29, 30, 37, 42, 48, 50, 61, 67, 77, 86, 99, 110, 122, 132, 153, 166, 188, 199, 226, 250, 268, 287, 330, 351, 379, 403, 450, 479, 536, 545, 610, 647, 691, 736, 821, 850, 918, 962, 1061, 1090, 1207, 1238, 1353, 1443, 1534, 1574, 1743, 1796, 1948, 2005, 2184, 2277, 2452.  Both assume Pyramids can be of any height and neither of these are in the OEIS yet.  I'm using Python as well, though I must embarrassingly admit that I'm not sure what an MMA expert is, since all my google searches turn up in the Mixed Martial Arts.  I'm more than happy to share my code with you if you want, though I am sure lots of people here could write something much more efficient.

-David
nacind at wpunj.edu



________________________________
From: Nacin, David <NACIND at wpunj.edu>
Sent: Thursday, July 22, 2021 6:18 PM
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: Re: [seqfan] Triangles of sums

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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