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%I #32 Oct 04 2023 12:31:55
%S 4,5,4,3,4,3,4,4,5,4,5,4,4,6,5,6,5,6,5,7,6,4,5,5,4,4,5
%N Starting with the n-th shortest Cartesian line segment, a(n) is the minimal number of consecutive line segments required to make a simple polygon.
%C List the possible lengths of line segments achievable by connecting integral coordinates on a Cartesian grid. Starting from the n-th length, a(n) is the smallest number of consecutively greater lengths required to form a simple polygon with all vertices on integral Cartesian coordinates.
%H Gordon Hamilton, <a href="https://mathpickle.com/project/rootingfornasa/">Rooting for NASA</a>.
%e a(4) = 3 because i) the fourth, fifth and sixth lengths are sqrt(5), sqrt(8) and 3 and ii) a triangle can be created using edges with these three lengths.
%e a(5) = 4 because i) the fifth, sixth, seventh and eighth lengths are sqrt(8), 3, sqrt(10), sqrt(13) and ii) a quadrilateral can be created using edges with these four lengths and iii) the fifth, sixth and seventh lengths alone cannot create a simple polygon with integral Cartesian vertices.
%Y Cf. A001481 (List of the squares of possible line segment lengths with both endpoints integral Cartesian coordinates).
%K nonn,more
%O 1,1
%A _Gordon Hamilton_, Sep 28 2023
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