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%I #21 Feb 18 2019 02:09:53
%S 0,0,1,1,2,36,36,36,50,53,153,153,153,333,333,333,360
%N Queen Dido's puzzle (the founding of Carthage): a(n) is twice the maximal area of a polygon with 1) vertices on integral Cartesian coordinates, 2) no two edges parallel, and 3) all edge lengths less than or equal to n^2.
%C The sequence may increase when n is the sum of two squares (A001481).
%C An optimal polygon will always be convex. - _Gordon Hamilton_
%C For parity reasons, the edges of the maximal-area polygon are not always as long as possible. This is true for a(9) through a(12). - _Gordon Hamilton_
%C This puzzle sequence could be used when introducing students to slopes.
%C Are these values known to be optimal or are they conjectures? - _N. J. A. Sloane_, Mar 13 2015
%C These values have not been proved to be optimal.
%e a(4) = 2 because this triangle has area 1 (remember a(n) is twice the area):
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%e a(5) = a(6) = a(7) = 36 because of this polygon of area 18:
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%e a(8) = 50 because of this polygon of area 25:
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%e a(9) = 53 because of this polygon of area 26.5:
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%e a(10) = 153 because of this polygon of area 76.5:
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%K nonn,more
%O 0,5
%A _Gordon Hamilton_, Mar 13 2015
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