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

1,2

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.

Links

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

Formula

a(p) = 2 for prime p.

Example

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.

Sequence in context: A133631 A137450 A344998 * A319692 A308693 A339481

Adjacent sequences: A321412 A321413 A321414 * A321416 A321417 A321418

Keyword

nonn,more

Author

Andrew Howroyd, Nov 08 2018

Status

approved