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

0, 2, 2, 10, 2, 64, 2, 330, 1028, 2010, 2, 24216, 2, 77528, 964696, 490314, 2, 11437750, 2, 21390330

Offset: 1

Author

Andrew Howroyd, Nov 08 2018

Keywords

• nonn
• more

Comments

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.

Examples

			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
		

Crossrefs

Main diagonal of A321414.

Cf. A262181, A292355.

Formula

a(p) = 2 for prime p.