# [seqfan] New Sequence Help Request (Terms and Bounds)

Zach DeStefano zachdestefano at gmail.com
Fri Jan 21 23:43:14 CET 2022

```All,

I am interested in help with techniques for computing additional terms or
proving better upper and lower bounds (hopefully constructive) on a
sequence which I recently submitted. Here is a description of the sequence
and a few notes, observations, and examples.

Description:

Earlier this month, I submitted the sequence A350547, "Maximum size of a
set of points taken from a hexagonal section of a hexagonal grid with side
length n such that no three selected points form an equilateral triangle."
I have confirmed a few initial terms using a combination of
exhaustive searches and integer programing: "1, 4, 9, 15, 22, 28, 36," and
I have the following successive lower bounds from a variety of methods:
"44, 52, 60, 66, 74, 82, 94, 98, 106, 116, 126, 132, 136, 142." From that
list, I suspect the 44, 52, and 60 are optimal and the 94 is the next
closest to optimal.

Observations:

For all currently known values and lower bounds, there exists a maximum
size set of points with reflection symmetry.

The terms appear to grow like ~O(n^(4/3)); however, my best lower bounds
are all linear or sublinear and my best general upper bound is the trivial
O(n^2).

It is possible to get tight upper bounds on successive terms when prior
terms are known:
Let A003215 be h(n) and A350547 be s(n), then we have:
s(n) <= (s(n - 1) / h(n - 1)) * h(n).

Chains of adjacent selected points can never form a cycle.

Small Example:

Here is one s(6) = 36
. . . x x . x
x . . x . x . x
x . . x . . . . x
x . . x . . . . . .
. . . . . . . . . x x
x x . . . . . . . x . x
. . . . . . . . . . . . .
x x . . . . . . . x . x
. . . . . . . . . x x
x . . x . . . . . .
x . . x . . . . x
x . . x . x . x
. . . x x . x
There are a variety of interesting structures I've noted in larger lower
bounds, but nothing that I have been able to use for a superlinear lower
bound construction.

Additionally, once the sequence is accepted or rejected, I am planning to
put a webpage online with examples of all known lower bounds.

Thanks,
Zach

```