A372477 Areas of alternating equilateral and non-equilateral triangles that make up a three-leaf tiling over a regular triangular grid.

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 16, 17, 19, 20, 21, 24, 25, 26, 27, 28, 30, 31, 33, 34, 36, 37, 39, 42, 43, 44, 46, 48, 49, 50, 52, 56, 57, 60, 61, 63, 64, 65, 67, 70, 72, 73, 75, 76, 79, 81, 82, 84, 85, 86, 89, 90, 91, 93, 94, 97, 100

Offset: 1

Author

Yury Kazakov, S. P. Obukhov, Sean Sun, and N. A. Shikhova, May 02 2024

Keywords

• nonn

Comments

The plane is divided into three equal 120-degree slices, and the illustration depicts a single slice. Along the boundaries of the slice, we build equilateral triangles beginning from the center with areas of 1,4,9,16,25 and so on. Then we build a segment connecting the free neighboring vertices of the initial triangles; this segment becomes the base of a new equilateral triangle, etc. In the Picture 1 and Picture 2 presented in the links below, blue triangles are equilateral, white triangles are not equilateral. In accordance with Pick's theorem for triangle grids, all triangles have integer areas.

It is convenient to represent the sequence as a sequence of values of the areas of triangles located inside the slice of the plane. The first layer contains two triangles with areas (1,1) and is located in the center of the slice, the second layer contains triangles with areas (4,2,3,2,4), the third contains triangles with areas (9,6,7,5,7,6,9) and so on (Picture 1). Layer number n contains 2n+1 triangles (except for the first layer, which has 2 triangles). The sequence represents the values of the areas of these triangles collected into one set, with duplicate elements removed, sorted in increasing order.

Examples

			For L=4:
Number Layer n = 1, Min Layer 1, [1, 1]
Number Layer n = 2, Min Layer 2, [4, 2, 3, 2, 4]
Number Layer n = 3, Min Layer 5, [9, 6, 7, 5, 7, 6, 9]
Number Layer n = 4, Min Layer 10, [16, 12, 13, 10, 12, 10, 13, 12, 16]
Number Layer n = 5, Min Layer 16, [25, 20, 21, 17, 19, 16, 19, 17, 21, 20, 25]
Number of terms below L^2+1=17 is 12.
In increasing order, without duplicates: [1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 16].
Terms below 17 are a(1)=1, a(2)=2, ..., a(11)=13, a(12)=16.
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=== Alternative layout idea ===
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The table below lists the numbers in layer n for n = 1..5. For each layer n >= 2, the table shows a pair of rows; the upper and lower rows in each pair list the triangle areas computed using the above formulas for b(n,k) and c(n,k), respectively.
                                                              min.
  --+-----+-------+-------+-------+-------+-------+-------+  number
  n | b/c | k = 0 |   1   |   2   |   3   |   4   |   5   | in layer
  ==+=====+=======+=======+=======+=======+=======+=======+==========
  1 |  -  |  1  1 |       |       |       |       |       |     1
  --+-----+-------+-------+-------+-------+-------+-------+----------
  2 |  c  |     2 |     2 |       |       |       |       |     2
    |  b  |  4    |  3    |  4    |       |       |       |
  --+-----+-------+-------+-------+-------+-------+-------+----------
  3 |  c  |     6 |     5 |     6 |       |       |       |     5
    |  b  |  9    |  7    |  7    |  9    |       |       |
  --+-----+-------+-------+-------+-------+-------+-------+----------
  4 |  c  |    12 |    10 |    10 |    12 |       |       |    10
    |  b  | 16    | 13    | 12    | 13    | 16    |       |
  --+-----+-------+-------+-------+-------+-------+-------+----------
  5 |  c  |    20 |    17 |    16 |    17 |    20 |       |    16
    |  b  | 25    | 21    | 19    | 19    | 21    | 25    |
  --+-----+-------+-------+-------+-------+-------+-------+----------
		

Links

• Yury Kazakov, Table of n, a(n) for n = 1..1000
• N. A. Shikhova, Discussion of a problem on the social network VK (in Russian).
• N. A. Shikhova, Picture 1. The sequence single slice.
• N. A. Shikhova, Pick's theorem for triangular grid (in Russian).
• Yury Kazakov, Picture 2. The sequence of plane tiling.
• Wikipedia, Pick's theorem.

Crossrefs

Cf. A372498 (complement).

Cf. A003136, A001859 (min terms of layers).

Programs

Python

import math
L=10 #generates terms below L**2+1
Lmax=math.trunc((1+2*math.sqrt(3*L**2+1))/3)+1
tr=set()
tr.add(1)
for n in range(2,Lmax):
  for k in range(0,n):
    p1=n*n+k*k-k*n
    p2=p1+k-n
    if p1<=L**2:
      tr.add(p1)
    if p2<=L**2:
      tr.add(p2)
print('Number terms below', L**2+1, 'is', len(tr))
print(sorted(tr))

Formula

Terms in {a(n)} <= L^2 are computed as follows:

Let Lmax = floor((2*sqrt(3*L^2+1)+1)/3)+1;

for n=1..Lmax, compute the terms in layer n, which are

[b(n,0), c(n,0), b(n,1), c(n,1), ..., b(n,n-1), c(n,n-1), b(n,n)],

using the formulas

b(n,k) = n*n + k*k - k*n for k = 0..n

and

c(n,k) = n*n + k*k - k*n + k - n for k = 0..n-1;

sort terms b(n,k) <= L^2 and c(n,k) <= L^2 in increasing order, and remove duplicates.