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%I #43 Jan 23 2015 11:15:23
%S 0,5,1,3,2,4,0,5,2,1,4,3
%N Integer side lengths in arithmetic progression of simple convex hexagons with equal interior angles. Sequence gives the values of m for sides of lengths t+m*d, counterclockwise, for the two primitive solutions.
%C Hexagons, such as (2,6,4,5,3,7), taken counterclockwise, can also be written (2,7,3,5,4,6) if considered clockwise or rotated 180 degrees and still read counterclockwise.
%C Conjecture: hexagons are the only simple convex polygons with equal interior angles with such property, due to the fact that cos(Pi/3) = 1/2.
%C The smaller such hexagon with all prime length sides is (7, 157, 67, 37, 127, 97). The smaller area of the two is sqrt(3)(6t^2 + 30td + 29d^2)/4 and the greater is sqrt(3)d^2/2 more.
%e If first term is t = 1 and common difference is d = 1, we get (1, 6, 2, 4, 3, 5) and (1, 6, 3, 2, 5, 4); two hexagons with equal interior angles and all sides with consecutive integer lengths.
%e If t = 5 and d = 6 we get (5, 35, 11, 23, 17, 29) and (5, 35, 17, 11, 29, 23).
%K nonn,fini
%O 1,2
%A _Robin Garcia_, Jul 31 2012
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