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%I #25 Sep 12 2021 08:43:33
%S 0,4,20,68,168,368,676,1184,1912,2944,4292,6152,8456,11484,15164,
%T 19624,24944,31508,39076,48212,58656,70672,84284,100192,117888,138100,
%U 160580,185796,213568,245008,279116,317424,359280,405124,454868,509264,567640,631988,701228,776032,855968,943260,1035844
%N Number of polygons formed when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.
%C 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 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 Like A306302(n) is appears only 3-gons and 4-gons are generated by the infinite lines.
%H Chai Wah Wu, <a href="/A344993/b344993.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 Scott R. Shannon, <a href="/A344993/a344993_1.gif">Image for n = 2</a>.
%H Scott R. Shannon, <a href="/A344993/a344993_2.gif">Image for n = 3</a>.
%H Scott R. Shannon, <a href="/A344993/a344993_3.gif">Image for n = 4</a>.
%H Scott R. Shannon, <a href="/A344993/a344993_4.gif">Image for n = 5</a>.
%H Scott R. Shannon, <a href="/A344993/a344993_7.gif">Image for n = 6</a>.
%H Scott R. Shannon, <a href="/A344993/a344993_8.gif">Image for n = 7</a>.
%F 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 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 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 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 See the linked images for further examples.
%o (Python)
%o from sympy import totient
%o 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 See A347750 and A347751 for the numbers of vertices and edges in the finite part of the corresponding graph.
%Y Cf. A332612 (half the number of polygons outside the rectangles), A306302 (number of polygons inside the rectangles), A255843.
%K nonn
%O 0,2
%A _Scott R. Shannon_ and _N. J. A. Sloane_, Jun 05 2021
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