A164896 -id:A164896 - cp's OEIS frontend

A110981 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).

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

Offset: 1

Max Alekseyev, Jan 20 2008

nonn

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^(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 dThe 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)

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

Cf. A001037, A103314, A107847, A164896.

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

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

A273096 Number of rotationally inequivalent minimal relations of roots of unity of weight n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 3, 3, 4, 6, 18, 69

Offset: 0

Christopher E. Thompson, May 15 2016

In this context, a relation of weight n is a multiset of n roots of unity which sum to zero, and it is minimal if no proper nonempty sub-multiset sums to zero. Relations are rotationally equivalent if they are obtained by multiplying each element by a common root of unity.

Mann classified the minimal relations up to weight 7, Conway and Jones up to weight 9, and Poonen and Rubinstein up to weight 12.

			Writing e(x) = exp(2*Pi*i*x), then e(1/6)+e(1/5)+e(2/5)+e(3/5)+e(4/5)+e(5/6) = 0 and this is the unique (up to rotation) minimal relation of weight 6.

J. H. Conway and A. J. Jones, Trigonometric diophantine equations (On vanishing sums of roots of unity), Acta Arithmetica 30(3), 229-240 (1976).
Henry B. Mann, On linear relations between roots of unity, Mathematika 12(2), 107-117 (1965).
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Math. 11(1), 135-156 (1998). Also at arXiv:math/9508209 [math.MG] with some typos corrected.

Cf. A103314, A110981, A164896.

A361635 Number of strictly-convex unit-sided polygons with all internal angles equal to a multiple of Pi/n, ignoring rotational and reflectional copies.

Original entry on oeis.org

0, 1, 3, 4, 7, 16, 17, 28, 70, 85, 125, 392, 379, 704, 3359, 2248, 4111, 18510, 14309, 30820

Offset: 1

Roman Mecholsky, Mar 18 2023

			For n=3, a(3) is computed as follows:  The base angle is Pi/3 (60 degrees).  Thus any internal angle can only be either Pi/3 or 2*Pi/3.  Call an interior angle with Pi/3 a "1" and with 2*Pi/3 a "2".  Since all external angles will add to 2*Pi, we know that the only possible sequences (ignoring rotation and reflection) are {{1, 1, 1}, {1, 1, 2, 2}, {1, 2, 1, 2}, {1, 2, 2, 2, 2}, {2, 2, 2, 2, 2, 2}}.  However, neither {1, 1, 2, 2} nor {1, 2, 2, 2, 2} forms a closed polygon.  Thus the final set is {{1, 1, 1}, {1, 2, 1, 2}, {2, 2, 2, 2, 2, 2}}, which gives a(3) = 3.

Cf. A164896, A361659.

a(p) = (2^(p-1)-1)/p + 2^((p-1)/2) for odd prime p. - Andrew Howroyd, Mar 22 2023

a(7) and a(9) corrected and a(11)-a(20) from Andrew Howroyd, Mar 22 2023

A361659 Number of strictly convex unit-sided polygons with all internal angles equal to a multiple of Pi/n, treating polygons that have a unique mirror image as distinct but ignoring rotational copies.

Original entry on oeis.org

0, 1, 3, 4, 7, 17, 19, 34, 92, 115, 187, 616, 631, 1201, 6067, 4114, 7711, 35322, 27595, 59704, 328833, 190933, 364723, 2435778, 1579882, 2582059, 21013768, 9894292, 18512791, 377367013, 69273667, 134219794, 1678410949, 505301839, 1339499035, 14843799550

Offset: 1

Roman Mecholsky, Mar 19 2023

nonn

Cf. A164896, A262181, A361635 (up to rotations and reflections).

a(n) = A164896(2*n) - 2. - Andrew Howroyd, Mar 22 2023

a(7) corrected and terms a(9) and beyond from Andrew Howroyd, Mar 22 2023

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cp's OEIS Frontend

A110981 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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A273096 Number of rotationally inequivalent minimal relations of roots of unity of weight n.

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A361635 Number of strictly-convex unit-sided polygons with all internal angles equal to a multiple of Pi/n, ignoring rotational and reflectional copies.

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Author

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Examples

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Formula

Extensions

A361659 Number of strictly convex unit-sided polygons with all internal angles equal to a multiple of Pi/n, treating polygons that have a unique mirror image as distinct but ignoring rotational copies.

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Author

Keywords

Crossrefs

Formula

Extensions