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A321415 Number of n element multisets of the 2n-th roots of unity with zero sum. 2

%I

%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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Last modified February 25 08:33 EST 2021. Contains 341606 sequences. (Running on oeis4.)