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A383082 The number of distinct straightedge-and-compass constructions that can be made with a total of n lines and circles.

%I A383082 #26 May 08 2025 08:59:06
%S A383082 1,3,3,16,205,5886,542983
%N A383082 The number of distinct straightedge-and-compass constructions that can be made with a total of n lines and circles.
%C A383082 A straightedge-and-compass construction starts with 2 points marked on the plane, traditionally (0,0) and (1,0). One can use a straightedge to draw a line between two marked points or a compass to draw a circle centered at a marked point through another marked point.
%C A383082 New points occur at the intersections of lines or circles with lines or circles.
%C A383082 Constructions are considered distinct if the location of the lines, circles, and points is different. In particular, a translation of a construction is considered distinct. - _Peter Kagey_, May 07 2025
%H A383082 Peter Kagey, <a href="/A383082/a383082.pdf">Illustration of a(3)=16, with the origin marked as a red point</a>.
%H A383082 Peter Kagey, <a href="/A383082/a383082_1.pdf">Illustration of a(4)=205, with the origin marked as a red point</a>.
%H A383082 Wikipedia, <a href="https://en.wikipedia.org/wiki/Straightedge_and_compass_construction">Straightedge and compass construction</a>
%e A383082 For n = 0, the a(0) = 1 diagram is the one consisting of two points.
%e A383082 For n = 1, there are a(1) = 3 possible constructions:
%e A383082 1) a line between (0,0), and (1,0),
%e A383082 2) a circle of radius 1 centered at (0,0),
%e A383082 3) a circle of radius 1 centered at (1,0).
%e A383082 For n = 2, there are a(2) = 3 possible constructions:
%e A383082 1) a line between (0,0), and (1,0) and a circle of radius 1 centered at (0,0), which marks the point (-1,0);
%e A383082 2) a line between (0,0), and (1,0) and a circle of radius 1 centered at (1,0), which marks the point (2,0);
%e A383082 3) two circles, both of radius 1, centered at (0,0) and (1,0), which marks the points (1/2,sqrt(3/4)) and (1/2,-sqrt(3/4)).
%e A383082 For n=3, see the a(3)=16 diagrams in the link.
%Y A383082 Cf. A333944, A383083.
%K A383082 hard,nonn,more
%O A383082 0,2
%A A383082 _Peter Kagey_, Apr 15 2025