A321414 -id:A321414 - cp's OEIS frontend

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

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.

A321416 Number of n element multisets of the 10th roots of unity with zero sum.

Original entry on oeis.org

1, 0, 5, 0, 15, 2, 35, 10, 70, 30, 128, 70, 220, 140, 360, 254, 565, 430, 855, 690, 1255, 1060, 1795, 1570, 2510, 2256, 3440, 3160, 4630, 4330, 6132, 5820, 8005, 7690, 10315, 10008, 13135, 12850, 16545, 16300, 20634, 20450, 25500, 25400, 31250, 31260

Offset: 0

Andrew Howroyd, Nov 09 2018

Equivalently, the number of closed convex paths of length n whose steps are the 10th roots of unity up to translation. For even n, there will be 5 paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,7,3,-8,3,7,-6,-4,4,1,-1)

Column k=5 of A321414.

Cf. A053090, A070190.

LinearRecurrence[{1, 4, -4, -6, 7, 3, -8, 3, 7, -6, -4, 4, 1, -1},{1, 0, 5, 0, 15, 2, 35, 10, 70, 30, 128, 70, 220, 140}, 50] (* Jinyuan Wang, Feb 28 2020 *)

Vec((1 - x^10)/((1 - x^2)^5 * (1 - x^5)^2) + O(x^50))

G.f.: (1 - x^10)/((1 - x^2)^5 * (1 - x^5)^2).

G.f.: (1 - x + x^2 - x^3 + x^4)/((1 + x + x^2 + x^3 + x^4)*(1 - x)^6*(1 + x)^4).

A321417 Number of n element multisets of the 12th roots of unity with zero sum.

Original entry on oeis.org

1, 0, 6, 4, 21, 24, 64, 84, 174, 236, 420, 576, 926, 1260, 1896, 2540, 3639, 4800, 6618, 8592, 11499, 14700, 19200, 24204, 30972, 38544, 48480, 59620, 73884, 89892, 109960, 132480, 160221, 191308, 229038, 271248, 321809, 378264, 445128, 519608, 606954, 704016

Offset: 0

Andrew Howroyd, Nov 09 2018

Equivalently, the number of closed convex paths of length n whose steps are the 12th roots of unity up to translation. For even n, there will be 6 paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-6,-6, 6,13,-2,-18, -2,13,6,-6, -6,3,2,-1)

Column k=6 of A321414.

Cf. A053090, A198808.

Vec(((2/(1 - x^3) - 1)/(1 - x^2)^3)^2 + O(x^40))

G.f.: ((2/(1 - x^3) - 1)/(1 - x^2)^3)^2.

G.f.: (1 - x + x^2)^2/((1 + x + x^2)^2*(1 - x)^8*(1 + x)^4).

A321419 Number of n element multisets of the 30th roots of unity with zero sum.

Original entry on oeis.org

1, 0, 15, 10, 120, 156, 760, 1350, 4350, 8770, 22896, 47610, 110110, 226260, 483180, 964696

Offset: 0

Andrew Howroyd, Nov 09 2018

Equivalently, the number of closed convex paths of length n whose steps are the 30th roots of unity up to translation. For even n, there will be 15 paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.

Column k=15 of A321414.

Cf. A321418.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A321416 Number of n element multisets of the 10th roots of unity with zero sum.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A321417 Number of n element multisets of the 12th roots of unity with zero sum.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A321419 Number of n element multisets of the 30th roots of unity with zero sum.

Views

Author

Keywords

Comments

Crossrefs