A344993 Number of polygons formed when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.
0, 4, 20, 68, 168, 368, 676, 1184, 1912, 2944, 4292, 6152, 8456, 11484, 15164, 19624, 24944, 31508, 39076, 48212, 58656, 70672, 84284, 100192, 117888, 138100, 160580, 185796, 213568, 245008, 279116, 317424, 359280, 405124, 454868, 509264, 567640, 631988, 701228, 776032, 855968, 943260, 1035844
Offset: 0
Keywords
Examples
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. 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. See the linked images for further examples.
Links
- Chai Wah Wu, Table of n, a(n) for n = 0..10000
- Scott R. Shannon, Image for n = 1. 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.
- Scott R. Shannon, Image for n = 2.
- Scott R. Shannon, Image for n = 3.
- Scott R. Shannon, Image for n = 4.
- Scott R. Shannon, Image for n = 5.
- Scott R. Shannon, Image for n = 6.
- Scott R. Shannon, Image for n = 7.
Crossrefs
Programs
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Python
from sympy import totient 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
Formula
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.
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
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