cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

This page as a plain text file.
%I A321415 #10 Nov 09 2018 21:33:37
%S A321415 0,2,2,10,2,64,2,330,1028,2010,2,24216,2,77528,964696,490314,2,
%T A321415 11437750,2,21390330
%N A321415 Number of n element multisets of the 2n-th roots of unity with zero sum.
%C A321415 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 A321415 Compared with A262181, this sequence counts all rotations distinctly and also for even n includes n/2 flat polygons as described above.
%C A321415 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 A321415 a(p) = 2 for prime p.
%e A321415 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 A321415                o---o    o---o
%e A321415   o===o===o    |   |     \   \
%e A321415                o---o      o---o
%Y A321415 Main diagonal of A321414.
%Y A321415 Cf. A262181, A292355.
%K A321415 nonn,more
%O A321415 1,2
%A A321415 _Andrew Howroyd_, Nov 08 2018