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 A344993 #25 Sep 12 2021 08:43:33
%S A344993 0,4,20,68,168,368,676,1184,1912,2944,4292,6152,8456,11484,15164,
%T A344993 19624,24944,31508,39076,48212,58656,70672,84284,100192,117888,138100,
%U A344993 160580,185796,213568,245008,279116,317424,359280,405124,454868,509264,567640,631988,701228,776032,855968,943260,1035844
%N A344993 Number of polygons formed when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.
%C A344993 The number of polygons formed inside the rectangles is A306302(n), while the number of polygons formed outside the rectangles is 2*A332612(n+1).
%C A344993 The number of open regions, those outside the polygons with unbounded area and two edges that go to infinity, for n >= 1 is given by 2*n^2 + 4*n + 6 = A255843(n+1).
%C A344993 Like A306302(n) is appears only 3-gons and 4-gons are generated by the infinite lines.
%H A344993 Chai Wah Wu, <a href="/A344993/b344993.txt">Table of n, a(n) for n = 0..10000</a>
%H A344993 Scott R. Shannon, <a href="/A344993/a344993.gif">Image for n = 1</a>. In this and other images the vertices at the corners of all rectangles are highlighted as white dots while the outer open regions, which are not counted, are darkened.
%H A344993 Scott R. Shannon, <a href="/A344993/a344993_1.gif">Image for n = 2</a>.
%H A344993 Scott R. Shannon, <a href="/A344993/a344993_2.gif">Image for n = 3</a>.
%H A344993 Scott R. Shannon, <a href="/A344993/a344993_3.gif">Image for n = 4</a>.
%H A344993 Scott R. Shannon, <a href="/A344993/a344993_4.gif">Image for n = 5</a>.
%H A344993 Scott R. Shannon, <a href="/A344993/a344993_7.gif">Image for n = 6</a>.
%H A344993 Scott R. Shannon, <a href="/A344993/a344993_8.gif">Image for n = 7</a>.
%F A344993 a(n) = 2*A332612(n+1) + A306302(n) = 2*Sum_{i=2..n, j=1..i-1, gcd(i,j)=1} (n+1-i)*(n+1-j) + Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) + n^2 + 2*n.
%F A344993 a(n) = 2*n*(n+1) + 2*Sum_{i=2..n} (n+1-i)*(2*n+2-i)*phi(i). - _Chai Wah Wu_, Aug 21 2021
%e A344993 a(1) = 4 as connecting the four vertices of a single rectangle forms four triangles inside the rectangle. Twelve open regions outside these triangles are also formed.
%e A344993 a(2) = 20 as connecting the six vertices of two adjacent rectangles forms two quadrilaterals and fourteen triangles inside the rectangles while also forming four triangles outside the rectangles, giving twenty polygons in total. Twenty-two open regions outside these polygons are also formed.
%e A344993 See the linked images for further examples.
%o A344993 (Python)
%o A344993 from sympy import totient
%o A344993 def A344993(n): return 2*n*(n+1) + 2*sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(2,n+1)) # _Chai Wah Wu_, Aug 21 2021
%Y A344993 See A347750 and A347751 for the numbers of vertices and edges in the finite part of the corresponding graph.
%Y A344993 Cf. A332612 (half the number of polygons outside the rectangles), A306302 (number of polygons inside the rectangles), A255843.
%K A344993 nonn
%O A344993 0,2
%A A344993 _Scott R. Shannon_ and _N. J. A. Sloane_, Jun 05 2021