

A110981


a(n) = the number of aperiodic subsets S of the nth roots of 1 with zero sum (i.e., there is no r different from 1 such that r*S=S).


5



1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 24, 0, 6, 0, 0, 0, 236, 0, 0, 0, 18, 0, 3768, 0, 0, 0, 0, 0, 20384, 0, 0, 0, 7188, 0, 227784, 0, 186, 480, 0, 0, 1732448, 0, 237600, 0, 630, 0, 16028160, 0, 306684, 0, 0, 0, 341521732, 0, 0, 4896, 0, 0, 1417919208
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OFFSET

1,12


COMMENTS

We count these subsets only modulo rotations (multiplication by a nontrivial root of unity).
A103314(n) = a(n)*n + 2^n  A001037(n)*n. Note that as soon as a(n)=0, we have simply A103314(n) = 2^n  A001037(n)*n. This makes it especially interesting to study those n for which a(n)=0. Quite surprisingly, it appears that the sequence of such n coincides with A102466.
From Max Alekseyev, Jan 31 2008: (Start)
Every subset of the set U(n) = { 1=r^0, r^1, ..., r^(n1) } of the nth power roots of 1 (where r is a fixed primitive root) defines a binary word w of the length n where the jth bit is 1 iff the root r^j is included in the subset.
If d is the period of w with respect to cyclic rotations (thus dn) then the periodic part of w uniquely defines some binary Lyndon word of the length d (see A001037). In turn, each binary Lyndon word of the length d, where d<n and dn, corresponds to d distinct zerosum subsets of U(n).
The binary Lyndon words of the length n are different in this respect: only some of them correspond to n distinct zerosum subsets of U(n) while the others do not correspond to such subsets at all. A110981(n) gives the number of binary Lyndon words of the length n that correspond to zerosum subsets of U(n). (End)


LINKS

Max Alekseyev and M. F. Hasler, Table of n, a(n) for n = 1..164


FORMULA

a(n) = A001037(n)  A107847(n) ( = A001037(n)  (2^n  A103314(n))/n ).  M. F. Hasler, Jan 31 2008


CROSSREFS

Cf. A001037, A103314, A107847, A164896.
Sequence in context: A189877 A189868 A336104 * A019261 A019222 A019141
Adjacent sequences: A110978 A110979 A110980 * A110982 A110983 A110984


KEYWORD

nonn


AUTHOR

Max Alekseyev, Jan 20 2008


EXTENSIONS

Additional comments from M. F. Hasler, Jan 31 2008


STATUS

approved



