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%I #10 Nov 09 2018 21:33:37
%S 0,2,2,10,2,64,2,330,1028,2010,2,24216,2,77528,964696,490314,2,
%T 11437750,2,21390330
%N Number of n element multisets of the 2n-th roots of unity with zero sum.
%C Equivalently, the number of closed convex paths of length n whose steps are the 2n-th roots of unity up to translation. For even n, there will be n paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.
%C Compared with A262181, this sequence counts all rotations distinctly and also for even n includes n/2 flat polygons as described above.
%C For prime n, a(n) is always 2. For odd prime the two solutions are the one consisting of all n-th roots of unity and the other consisting of their negated values.
%F a(p) = 2 for prime p.
%e a(4) = 12 because there are 3 basic shapes illustrated below which with rotations of multiples of 45 degrees give 4 + 2 + 4 = 10 distinct convex paths.
%e o---o o---o
%e o===o===o | | \ \
%e o---o o---o
%Y Main diagonal of A321414.
%Y Cf. A262181, A292355.
%K nonn,more
%O 1,2
%A _Andrew Howroyd_, Nov 08 2018
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