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A110981
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a(n) = the number of aperiodic subsets S of the n-th roots of 1 with zero sum (i.e., there is no r different from 1 such that r*S=S).
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5
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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
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1,12
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COMMENTS
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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.
Every subset of the set U(n) = { 1=r^0, r^1, ..., r^(n-1) } of the n-th power roots of 1 (where r is a fixed primitive root) defines a binary word w of the length n where the j-th 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 d|n) 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 d|n, corresponds to d distinct zero-sum 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 zero-sum 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 zero-sum subsets of U(n). (End)
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LINKS
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FORMULA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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