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 A324042 #57 Aug 16 2021 13:47:05
%S A324042 4,14,32,70,124,226,360,566,820,1218,1696,2310,3020,4018,5160,6590,
%T A324042 8196,10218,12464,15110,18012,21650,25624,30142,35028,40954,47344,
%U A324042 54558,62284,71034,80360,90806,101892,114770,128416,143286,158972,176914,195816,216350,237908,261546,286304,313102,341100
%N A324042 Number of triangular regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles.
%C A324042 A row of n adjacent congruent rectangles can only be divided into triangles or quadrilaterals when drawing diagonals. A proof is given in Alekseyev et al. (2015) using the mapping to a dissection of a a right isosceles triangle described in A306302.
%H A324042 Chai Wah Wu, <a href="/A324042/b324042.txt">Table of n, a(n) for n = 1..10000</a>
%H A324042 M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. <a href="https://doi.org/10.1137/140978090">On the minimal teaching sets of two-dimensional threshold functions</a>. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165.
%H A324042 Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, <a href="http://neilsloane.com/doc/rose_5.pdf">Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids</a>, (2021). Also arXiv:2009.07918.
%H A324042 Robert Israel, <a href="/A324042/a324042.txt">Maple program</a>
%H A324042 Jinyuan Wang, <a href="/A324042/a324042.png">Illustration for n = 1, 2, 3, 4, 5</a>
%F A324042 a(n) = A177719(n+1) + 2*(n+1) = 2 * ( (n+1)*n + Sum_{i,j=1..n; gcd(i,j)=2} (n+1-i)*(n+1-j) ). - _Max Alekseyev_, Jul 08 2019
%F A324042 a(n) = A306302(n) - A324043(n).
%F A324042 a(n) = 2*(2*n^2-n+1+2*Sum_{i=2..floor(n/2)} (n+1-2*i)*(n+1-i)*phi(i)). - _Chai Wah Wu_, Aug 16 2021
%e A324042 For k adjacent congruent rectangles, the number of triangular regions in the j-th rectangle is:
%e A324042 k\j| 1 2 3 4 5 6 7 ...
%e A324042 ---+--------------------------------
%e A324042 1 | 4, 0, 0, 0, 0, 0, 0, ...
%e A324042 2 | 7, 7, 0, 0, 0, 0, 0, ...
%e A324042 3 | 9, 14, 9, 0, 0, 0, 0, ...
%e A324042 4 | 11, 24, 24, 11, 0, 0, 0, ...
%e A324042 5 | 13, 30, 38, 30, 13, 0, 0, ...
%e A324042 6 | 15, 38, 60, 60, 38, 15, 0, ...
%e A324042 7 | 17, 44, 76, 86, 76, 44, 17, ...
%e A324042 ...
%e A324042 a(4) = 11 + 24 + 24 + 11 = 70.
%p A324042 V := proc(m,n,q) local a,i,j; a:=0;
%p A324042 for i from 1 to m do for j from 1 to n do
%p A324042 if gcd(i,j)=q then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
%p A324042 a := n -> 2*( n*(n+1) + V(n,n,2) );
%p A324042 [seq(a(n), n=1..30)]; # _N. J. A. Sloane_, Mar 04 2020
%p A324042 See also Robert Israel link.
%t A324042 Table[2 * (n^2 + n + Sum[Sum[Boole[GCD[i, j] == 2] * (n + 1 - i) * (n + 1 - j), {j, 1, n}], {i, 1, n}]), {n, 1, 45}] (* _Joshua Oliver_, Feb 05 2020 *)
%o A324042 (PARI) { A324042(n) = 2*((n+1)*n + sum(i=1, n, sum(j=1, n, (gcd(i, j)==2)*(n+1-i)*(n+1-j))) ); } \\ _Max Alekseyev_, Jul 08 2019
%o A324042 (Python)
%o A324042 from sympy import totient
%o A324042 def A324042(n): return 2*(2*n**2-n+1 + 2*sum(totient(i)*(n+1-2*i)*(n+1-i) for i in range(2,n//2+1))) # _Chai Wah Wu_, Aug 16 2021
%Y A324042 Cf. A177719, A306302, A324043, A333286, A333287, A333288.
%K A324042 nonn
%O A324042 1,1
%A A324042 _Jinyuan Wang_, May 01 2019
%E A324042 a(8)-a(23) from _Robert Israel_, Jul 07 2019
%E A324042 Terms a(24) onward from _Max Alekseyev_, Jul 08 2019