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 A328526 #20 May 02 2020 05:02:46 %S A328526 1,12,75,249,543,1023,1746,2814,4293,6267,8868,12228,16464,21774, %T A328526 28176,35832,45066,56040,68931,84033,101307,120987,143574,169290, %U A328526 198222,230790,267117,307455,352437,402255,457182,517986,584454,656874,735708,821076,913860 %N A328526 Number of regions in an equilateral triangle "frame" of size n. %C A328526 A equilateral triangular "frame" of size n is formed from a triangular grid consisting of an outer edge of (n+1) points with the central grid of (n-5)*(n-6)/2 points removed. If n is less than 4 then no points or triangles are removed, and a(n) = A092867(n). From now on we assume n >= 4. %C A328526 If we focus on the triangles rather than the points, the frame consists of a grid of equilateral triangles with the central block of (n-3)^2 triangles removed. %C A328526 The resulting structure has an outer perimeter with 3*n points and an inner perimeter with 3*n-9 points, for a total of 6*n-9 perimeter points. The frame itself is the strip equilateral triangles pointing in alternate directions between the inner and outer perimeters such that the frame thickness equals the height of one triangle. %C A328526 Now join every pair of perimeter points, both inner and outer, by a line segment, provided the line remains inside the frame. The sequence gives the number of regions in the resulting figure. %C A328526 Like the square frame of A331776 only regions with 3 or 4 edges are formed. %H A328526 Lars Blomberg, <a href="/A328526/b328526.txt">Table of n, a(n) for n = 1..65</a> %H A328526 Scott R. Shannon, <a href="/A328526/a328526_3.png">Illustration for n=7</a>. %H A328526 Scott R. Shannon, <a href="/A328526/a328526.png">Illustration for n=4 with random distance-based coloring</a>. %H A328526 Scott R. Shannon, <a href="/A328526/a328526_1.png">Illustration for n=7 with random distance-based coloring</a>. %H A328526 Scott R. Shannon, <a href="/A328526/a328526_2.png">Illustration for n=11 with random distance-based coloring</a>. %Y A328526 Cf. A333030 (edges), A333031 (vertices), A333032 (3-gons), A333033 (4-gons), A331776 (square frame), A092867 (filled triangle). %K A328526 nonn %O A328526 1,2 %A A328526 _Scott R. Shannon_ and _N. J. A. Sloane_, Feb 24 2020 %E A328526 a(12) and beyond from _Lars Blomberg_, May 01 2020