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

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

Author

Andrew Howroyd, Nov 09 2018

Keywords

• nonn
• easy

Comments

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.

Links

• 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)

Crossrefs

Column k=5 of A321414.

Cf. A053090, A070190.

Programs

Mathematica

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 *)

PARI

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

Formula

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).