A335411 a(n) is the number of vertices formed by n-secting the angles of an equilateral triangle.

3, 7, 21, 25, 63, 67, 129, 133, 219, 199, 333, 337, 471, 475, 633, 637, 819, 823, 1029, 1009, 1263, 1267, 1521, 1525, 1803, 1807, 2109, 2113, 2439, 2419, 2793, 2797, 3171, 3175, 3573, 3577, 3999, 4003, 4449, 4429, 4923, 4927, 5421, 5425, 5943, 5947, 6489

Offset: 1

Author

Lars Blomberg, Jun 08 2020

Keywords

• nonn

Comments

See A277402 for illustrations.

Links

• Lars Blomberg, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A331782, A277402 (regions), A335412 (edges), A335413 (ngons).

Formula

Empirically for 12 < n < 500: a(n) = a(n-2) + a(n-10) - a(n-12) + 120.

Conjectures from Colin Barker, Jun 08 2020: (Start)

G.f.: x*(3 + 4*x + 11*x^2 + 24*x^4 + 24*x^6 + 24*x^8 - 24*x^9 + 45*x^10 + 20*x^11 - 11*x^12) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-10) - a(n-11) - a(n-12) + a(n-13) for n>13.

(End)

Colin Barker's recurrence conjecture holds for 13 < n <= 500. Lars Blomberg, Jun 12 2020

Empirical: a(2*k - 1) = 3*(4*k^2 - 6*k + 3), for k >= 1. - Ivan N. Ianakiev, Jul 15 2020