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A321415 Number of n element multisets of the 2n-th roots of unity with zero sum. 2
0, 2, 2, 10, 2, 64, 2, 330, 1028, 2010, 2, 24216, 2, 77528, 964696, 490314, 2, 11437750, 2, 21390330 (list; graph; refs; listen; history; text; internal format)



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.

Compared with A262181, this sequence counts all rotations distinctly and also for even n includes n/2 flat polygons as described above.

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.


Table of n, a(n) for n=1..20.


a(p) = 2 for prime p.


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.

               o---o    o---o

  o===o===o    |   |     \   \

               o---o      o---o


Main diagonal of A321414.

Cf. A262181, A292355.

Sequence in context: A319880 A133631 A137450 * A319692 A308693 A163937

Adjacent sequences:  A321412 A321413 A321414 * A321416 A321417 A321418




Andrew Howroyd, Nov 08 2018



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Last modified January 24 01:20 EST 2021. Contains 340398 sequences. (Running on oeis4.)