A390763 a(n) is the number of 5 element sets of distinct integer-sided trapezoids each of area less than 3*n^2 whose base angles are 60 degrees that fill a regular hexagon of side n units.

0, 0, 0, 6, 20, 50, 96, 168, 264, 396, 560, 770, 1020, 1326, 1680, 2100, 2576, 3128, 3744, 4446, 5220, 5880, 7040, 8096, 9240, 10500, 11856, 13338, 14924, 16646, 18480, 20460, 22560, 24816, 27200, 29750, 32436, 35298, 38304, 41496, 44840, 48380, 52080, 55986, 60060, 64350, 68816

Offset

1,4

Comments

A trapezoid whose base angles are 60 degrees with larger base b and legs s is denoted by {b X s} here.

Starting with a hexagon ABCDEF partitioned into 2 identical trapezoids ABCF and FCDE, then splitting ABCF parallel to FC into 2 trapezoids, there are two mutually exclusive ways of obtaining a hexagon partitioned into 5 trapezoids each of area less than 3*n^2 in unit equilateral triangles.

Category A: Split the trapezoid FCDE parallel to FC into 3 trapezoids.

Category B: Split the trapezoid FCDE parallel to CD into a trapezoid and a parallelogram, then split the parallelogram parallel to BC into 2 trapezoids.

Links

Table of n, a(n) for n=1..47.

Formula

a(n) = p(n) + q(n) where:

Category A: p(n) = binomial(n-1,2)*(n-3),

Category B: q(n) = (number of sets belong to Category Y of A390739)*(n-1).

Example

For n = 4, there are 6 sets. Therefore a(4) = 6.
  {{5 X 1}, {8 X 1}, {7 X 1}, {6 X 2}, {8 X 3}},
  {{5 X 1}, {6 X 2}, {8 X 1}, {8 X 2}, {7 X 2}},
  {{5 X 1}, {6 X 1}, {7 X 3}, {8 X 1}, {8 X 2}},
  {{2 X 1}, {3 X 1}, {7 X 3}, {8 X 1}, {7 X 4}},
  {{2 X 1}, {3 X 1}, {6 X 2}, {7 X 4}, {8 X 2}},
  {{2 X 1}, {3 X 1}, {5 X 1}, {7 X 4}, {8 X 3}}.

Crossrefs

Cf. A390739, A389392.

Sequence in context: A161438 A063488 A299292 * A162209 A161699 A216175

Adjacent sequences: A390760 A390761 A390762 * A390764 A390765 A390766

Keyword

nonn

Author

Janaka Rodrigo, Nov 18 2025

Status

approved