A262181 -id:A262181 - cp's OEIS frontend

A321414 Array read by antidiagonals: T(n,k) is the number of n element multisets of the 2k-th roots of unity with zero sum.

0, 0, 1, 0, 2, 0, 0, 3, 0, 1, 0, 4, 2, 3, 0, 0, 5, 0, 6, 0, 1, 0, 6, 0, 10, 6, 4, 0, 0, 7, 4, 15, 0, 12, 0, 1, 0, 8, 0, 21, 2, 20, 12, 5, 0, 0, 9, 0, 28, 24, 35, 0, 21, 0, 1, 0, 10, 6, 36, 0, 64, 10, 35, 22, 6, 0, 0, 11, 0, 45, 0, 84, 84, 70, 0, 33, 0, 1

Offset: 1

Andrew Howroyd, Nov 08 2018

Equivalently, the number of closed convex paths of length n whose steps are the 2k-th roots of unity up to translation. For even n, there will be k paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.

			Array begins:
  =========================================================
  n\k| 1  2  3  4   5   6   7    8    9   10   11    12
  ---|-----------------------------------------------------
   1 | 0  0  0  0   0   0   0    0    0    0    0     0 ...
   2 | 1  2  3  4   5   6   7    8    9   10   11    12 ...
   3 | 0  0  2  0   0   4   0    0    6    0    0     8 ...
   4 | 1  3  6 10  15  21  28   36   45   55   66    78 ...
   5 | 0  0  6  0   2  24   0    0   54    4    0    96 ...
   6 | 1  4 12 20  35  64  84  120  183  220  286   396 ...
   7 | 0  0 12  0  10  84   2    0  270   40    0   624 ...
   8 | 1  5 21 35  70 174 210  330  657  715 1001  1749 ...
   9 | 0  0 22  0  30 236  14    0 1028  220    0  3000 ...
  10 | 1  6 33 56 128 420 462  792 2097 2010 3003  6864 ...
  11 | 0  0 36  0  70 576  56    0 3312  880    2 11976 ...
  12 | 1  7 50 84 220 926 924 1716 6039 5085 8008 24216 ...
  ...
T(5, 3) = 6 because there are 6 rotations of the following figure:
       o---o
      /     \
     o---o---o
.
T(6, 3) = 12 because there are 4 basic shapes illustrated below which with rotations and reflections give 3 + 2 + 1 + 6 = 12 convex paths.
                        o        o---o     o---o
                       / \      /     \     \   \
    o===o===o===o     o   o    o       o     o   o
                     /     \    \     /       \   \
                    o---o---o    o---o         o---o

Andrew Howroyd, Table of n, a(n) for n = 1..465

Main diagonal is A321415.
Columns include A053090(n+3), A321416, A321417, A321419.
Cf. A103306, A103314, A262181, A292355.

\\ only supports k with at most one odd prime factor.
T(n, k)={my(r=valuation(k, 2), p); polcoef(if(k>>r == 1, 1/(1-x^2)^k + O(x*x^n), if(isprimepower(k>>r, &p), ((2/(1 - x^p) - 1)/(1 - x^2 + O(x*x^n))^p)^(k/p), error("Cannot handle k=", k) )), n)}

G.f. of column k = 2^r: 1/(1 - x^2)^k - 1.

G.f. of column k = 2^r*p^e: ((2/(1 - x^p) - 1)/(1 - x^2)^p)^(k/p) - 1 for odd prime p.

A292355 Number of distinct convex equilateral n-gons having rotational symmetry and with corner angles of m*Pi/n (0 < m <= n).

Original entry on oeis.org

1, 2, 1, 11, 1, 42, 10, 202, 1, 1077, 1, 5539, 210, 30666, 1, 174620, 1, 1001642, 5547, 5864751, 1, 34799997, 201, 208267321, 173593, 1258579693, 1, 7664723137, 1, 46976034378, 5864759, 289628805624, 5738, 1794967236906, 1, 11175157356523, 208267329

Offset: 3

Andrew Howroyd, Sep 14 2017

nonn

Subset of polygons of A262181 having rotational symmetry. Polygons that differ only by rotation are not considered as distinct. See A262181 for illustrations of initial terms. The first difference between this sequence and A262181 is at a(9).

			Case n=6: The ways to select d angles that are multiples of Pi/n and sum to 2*d which are nonequivalent up to rotation and d is a proper factor of 6 are:
d = 1: {2}
d = 2: {04, 13}
d = 3: {015, 024, 033, 042, 051, 114, 123, 132}
In total there are 11 possibilities, so a(6) = 11.
In the above, 22 and 222 are excluded from the possibilities for d = 2 and 3 because they correspond to the regular hexagon that is covered by d = 1.
Also, 006 has been excluded from d = 3 since 6 corresponds to an angle of 180 degrees which is disallowed by this sequence. This would be the flattened polygon of three sides in one direction and then three back in the opposite.

Andrew Howroyd, Table of n, a(n) for n = 3..1000

Cf. A262181.

a(n) = -(1+(-1)^n)/2 + (1/n)*sumdiv(n,d, (eulerphi(n/d)-moebius(n/d)) * binomial(3*d-1, d-1));

a(n) = -(1+(-1)^n)/2 + (1/n)*Sum_{d | n} (phi(n/d)-moebius(n/d)) * binomial(3*d-1, d-1).
a(n) = A262181(n) for n prime or twice prime.
Conjecture: a(2^n) = A262181(2^n).

A262244 Number of concave equilateral n-gons with corner angles of m*Pi/n (0 < m < 2n), where m and n are integers.

Original entry on oeis.org

0, 0, 1, 22, 11, 1319, 25858

Offset: 3

Stuart E Anderson, Sep 15 2015

An n-gon is a polygon with n corners and n sides, each of which is a straight line segment joining two corners. A polygon P is said to be a simple polygon (or a Jordan polygon) if the only points of the plane belonging to multiple edges of P are the corners of P. Such a polygon has a well-defined interior and exterior. Simple polygons are topologically equivalent to a disk, hence zero angles are not allowed; allowable angles are m*Pi/n (where m and n are integers and 0 < m < 2n). A simple n-gon is concave iff at least one of its internal angles is greater than Pi, or equivalently m > n for at least one of the corners. The sum of the m-numbers (called angle factors) for the n-gon has to be n*(n-2). They are partitions of n*(n-2) into n parts with largest part n < k < 2n, and as the edges of a polygon form a closed path, the sum of unit vectors defined by the angle coordinates m/Pi is zero. The reason the m-numbers sum to n*(n-2) is that the sum of the interior angles of any n-gon is Pi*(n-2), and as angles are m*Pi/n, n = Pi.

Observation: when n is prime, m is odd and m != n.

			For n = 5, the a(5) = 1 solution is (1 3 3 1 7) in m angle factors.
For n = 7, the a(7) = 11 solutions in m angle factors are as follows: (1 11 5 3 5 5 5), (1 5 3 9 1 5 11), (1 5 5 1 11 1 11), (1 5 5 5 1 9 9), (1 5 5 5 3 5 11), (1 9 1 9 3 3 9), (1 9 3 5 1 11 5), (1 9 3 5 5 3 9), (3 3 5 5 3 3 13), (3 3 9 3 5 3 9), (3 5 5 5 5 3 9).

Stuart E Anderson, concave pentagon
Stuart E Anderson, concave hexagons
Stuart E Anderson, concave heptagons
Stuart E Anderson, concave octagons
Stuart E Anderson, jk-latest.cpp produces postscript images of convex, concave and intersecting polygons for n, alternatively there is a switch to produce an unsorted list of interior angle multiples m for each polygon. There is a switch to select if reflected polygons are counted. Polygons produced are all invariant under rotation, new version, Aug 04 2024.

Cf. A262181 (convex equilateral polygons).

a(9) corrected by Stuart E Anderson, Aug 04 2024

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

Original entry on oeis.org

0, 2, 2, 10, 2, 64, 2, 330, 1028, 2010, 2, 24216, 2, 77528, 964696, 490314, 2, 11437750, 2, 21390330

Offset: 1

Andrew Howroyd, Nov 08 2018

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.

			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

Main diagonal of A321414.

Cf. A262181, A292355.

a(p) = 2 for prime p.

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

cp's OEIS Frontend

A321414 Array read by antidiagonals: T(n,k) is the number of n element multisets of the 2k-th roots of unity with zero sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A292355 Number of distinct convex equilateral n-gons having rotational symmetry and with corner angles of m*Pi/n (0 < m <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A262244 Number of concave equilateral n-gons with corner angles of m*Pi/n (0 < m < 2n), where m and n are integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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.

Views

Author

Keywords

Crossrefs

Formula

Extensions