[seqfan] Nice question about covering triangles

[Neil Sloane]

Dear Seq Fans, A new contributor, Anthony Bartoletti, azb at comcast.net
wrote to me about two interesting sequences that he needs help with.  Would
someone on this list please help him?
(I said he should join this list so he can post messages himself in future)

Here is what he said:

I believe I have two sequences that are not listed, and I have
"established" the first 5 terms of each.  Unfortunately, I have developed
no algorithm to produce the values, which I have only found through manual
construction and enumeration.



These are the "distinct triangular covers of triangular arrays",
discounting and counting orientations.



                discounting:       1, 2, 4, 11, 63, ...

                counting:             1, 2, 6, 30, 253, ...



The attached ASCII text file provides the figures, annotated throughout
with my "strategy" and discussion for exhausting the possibilities.  The
cases for n = 1,2,3,4 are easy to dispatch.  The case n=5 is a bit more
challenging.



I'd like to add these, as they seem reasonably straightforward in concept
and foundation, but I have no idea how to craft entries for them, or point
anyone to the text file for examination.  Besides the text file, all that
remains are the numbers and the stated definitions.



I hope you can assist.  Really, there is no hurry at all - I'd prefer not
to impose upon your time.



Sincerely, Tony Bartoletti

and he attached a plain text  which I transcribe here:

Herein, I offer two related sequences, and argue for their correctness
pictorially.

a(n) = triangular covers of triangular arrays of edge size n, ignoring
orientations.
     = 1, 2, 4, 11, 63, ...

b(n) = triangular covers of triangular arrays of edge size n,
including distinguishable reflections and rotations.
     = 1, 2, 6, 30, 253, ...

Discussion:  To depict triangles in ASCII text, I chose to select a
different letter (A, B, C,...) to "paint" each triangle.

For example, the triangle of edge 5 at right             A
is covered by 6 triangles - one [A] of edge 3,          A A
two [B,C] of edge 2, and three [D,E,F] of edge 1.      A A A
Rather than call them by edges, I often call          B C C D
them by interior count, and number of copies.        B B C E F
Hence, here we cover a 15:1 with 6:1,3:2,1:3

Sometimes in the text, I will refer to a "3" by its edge count, as an
"e2" for readability.

In general, the order in which the covers are demonstrated involves
using the largest possible covers the most possible times.
Where this allows multiple configurations, I order by "number
occupying corners" and other tricks discussed below in context.
I believe these tricks convincingly exhaust the possible covers.
Where helpful, I indicate the orientations just below the array.

a(1) = 1 (oriented b(1) = 1)

	 A

	1:1


a(2) = 2 (oriented b(2) = 2)

	 A	 A
	A A	B C

	3:1	1:3


a(3) = 4 (oriented b(3) = 6)




	  A	  A	  B	  A
	 A A	 A A	 A A	 B C
	A A A	B C D	C A D	D E F

	  1	  3	  1	  1

	 6:1   3:1,1:3 3:1,1:3   1:6


a(4) = 11 (oriented b(4) = 30)

	   A	     A	       A	 D	   C	     A	       A	 A	   B	     B	       A
	  A A	    A A	      A A	A A	  D E	    A A	      A A	A A	  A A	    C D	      B C
	 A A A	   A A A     B D C     B A C	 A F B	   C B D     B B C     B C
D	 C A D	   E A F     D E F
	A A A A	  B C D E   B B C C   B B C C   A A B B   E B B F   D B E F
E F G H	E F G H	  G A A H   G H I J

	   1	     3	       1	 3	   3	     3	       6	 3	   3	     3	       1

	 10:1     6:1,1:4   3:3,1:1   3:3,1:1   3:2,1:4   3:2,1:4   3:2,1:4
3:1,1:7   3:1,1:7   3:1,1:7     1:10

      Figures 1, 2 and 11 above are trivial.  Figures 3 through 10
deal with arrangements of 3, 2 or just 1 triangle of edge 2 (e2).
      Figure 3 forces the 3 e2 into the corners, while figure 4 (2
corners) is the only remaining option for 3 of e2.
      Figures 5,6,7 provide all possible arrangements with 2 of e2,
one taking two corners, and two taking 1 corner.
      Likewise, 8,9,10 demonstrate all distinguishable placements of a
single e2 (corner, sub-corner, and edge).
      (I refer to the closest non-corner position symmetric to a
corner as the "sub-corner", such as triangle [A] in figure 4 above.)


a(5) = 63 (oriented b(5) = 253)

........


(I stop there - there is a lot more which I will send to anyone who is
interested in helping Tony)



Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com