This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372477 #117 Jun 22 2024 18:18:06
%S A372477 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,
%T A372477 36,37,39,42,43,44,46,48,49,50,52,56,57,60,61,63,64,65,67,70,72,73,75,
%U A372477 76,79,81,82,84,85,86,89,90,91,93,94,97,100
%N A372477 Areas of alternating equilateral and non-equilateral triangles that make up a three-leaf tiling over a regular triangular grid.
%C A372477 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.
%C A372477 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.
%H A372477 Yury Kazakov, <a href="/A372477/b372477.txt">Table of n, a(n) for n = 1..1000</a>
%H A372477 N. A. Shikhova, <a href="https://vk.com/wall-125266332_63183">Discussion of a problem on the social network VK</a> (in Russian).
%H A372477 N. A. Shikhova, <a href="https://drive.google.com/file/d/1MfQnTIX7Aw-Y5ygTrUjRFryFuRna7OPf/view?usp=drive_link">Picture 1. The sequence single slice</a>.
%H A372477 N. A. Shikhova, <a href="https://dzen.ru/a/XMDJg_GEOQCyuLrX">Pick's theorem for triangular grid</a> (in Russian).
%H A372477 Yury Kazakov, <a href="https://drive.google.com/file/d/1pLxAxqcIGjx0-W39Kriiu9HpecUbVpYa/view?usp=drive_link">Picture 2. The sequence of plane tiling</a>.
%H A372477 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pick%27s_theorem">Pick's theorem</a>.
%F A372477 Terms in {a(n)} <= L^2 are computed as follows:
%F A372477 Let Lmax = floor((2*sqrt(3*L^2+1)+1)/3)+1;
%F A372477 for n=1..Lmax, compute the terms in layer n, which are
%F A372477 [b(n,0), c(n,0), b(n,1), c(n,1), ..., b(n,n-1), c(n,n-1), b(n,n)],
%F A372477 using the formulas
%F A372477 b(n,k) = n*n + k*k - k*n for k = 0..n
%F A372477 and
%F A372477 c(n,k) = n*n + k*k - k*n + k - n for k = 0..n-1;
%F A372477 sort terms b(n,k) <= L^2 and c(n,k) <= L^2 in increasing order, and remove duplicates.
%e A372477 For L=4:
%e A372477 Number Layer n = 1, Min Layer 1, [1, 1]
%e A372477 Number Layer n = 2, Min Layer 2, [4, 2, 3, 2, 4]
%e A372477 Number Layer n = 3, Min Layer 5, [9, 6, 7, 5, 7, 6, 9]
%e A372477 Number Layer n = 4, Min Layer 10, [16, 12, 13, 10, 12, 10, 13, 12, 16]
%e A372477 Number Layer n = 5, Min Layer 16, [25, 20, 21, 17, 19, 16, 19, 17, 21, 20, 25]
%e A372477 Number of terms below L^2+1=17 is 12.
%e A372477 In increasing order, without duplicates: [1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 16].
%e A372477 Terms below 17 are a(1)=1, a(2)=2, ..., a(11)=13, a(12)=16.
%e A372477 .
%e A372477 ===============================
%e A372477 === Alternative layout idea ===
%e A372477 ===============================
%e A372477 .
%e A372477 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.
%e A372477 min.
%e A372477 --+-----+-------+-------+-------+-------+-------+-------+ number
%e A372477 n | b/c | k = 0 | 1 | 2 | 3 | 4 | 5 | in layer
%e A372477 ==+=====+=======+=======+=======+=======+=======+=======+==========
%e A372477 1 | - | 1 1 | | | | | | 1
%e A372477 --+-----+-------+-------+-------+-------+-------+-------+----------
%e A372477 2 | c | 2 | 2 | | | | | 2
%e A372477 | b | 4 | 3 | 4 | | | |
%e A372477 --+-----+-------+-------+-------+-------+-------+-------+----------
%e A372477 3 | c | 6 | 5 | 6 | | | | 5
%e A372477 | b | 9 | 7 | 7 | 9 | | |
%e A372477 --+-----+-------+-------+-------+-------+-------+-------+----------
%e A372477 4 | c | 12 | 10 | 10 | 12 | | | 10
%e A372477 | b | 16 | 13 | 12 | 13 | 16 | |
%e A372477 --+-----+-------+-------+-------+-------+-------+-------+----------
%e A372477 5 | c | 20 | 17 | 16 | 17 | 20 | | 16
%e A372477 | b | 25 | 21 | 19 | 19 | 21 | 25 |
%e A372477 --+-----+-------+-------+-------+-------+-------+-------+----------
%o A372477 (Python)
%o A372477 import math
%o A372477 L=10 #generates terms below L**2+1
%o A372477 Lmax=math.trunc((1+2*math.sqrt(3*L**2+1))/3)+1
%o A372477 tr=set()
%o A372477 tr.add(1)
%o A372477 for n in range(2,Lmax):
%o A372477 for k in range(0,n):
%o A372477 p1=n*n+k*k-k*n
%o A372477 p2=p1+k-n
%o A372477 if p1<=L**2:
%o A372477 tr.add(p1)
%o A372477 if p2<=L**2:
%o A372477 tr.add(p2)
%o A372477 print('Number terms below', L**2+1, 'is', len(tr))
%o A372477 print(sorted(tr))
%Y A372477 Cf. A372498 (complement).
%Y A372477 Cf. A003136, A001859 (min terms of layers).
%K A372477 nonn
%O A372477 1,2
%A A372477 _Yury Kazakov_, S. P. Obukhov, Sean Sun, and N. A. Shikhova, May 02 2024