A390739 a(n) is the least possible difference between the areas of the largest and smallest trapezoids in a 4 element set of distinct integer-sided trapezoids whose base angles are 60 degrees that fill a regular hexagon of side n units.

13, 18, 27, 20, 43, 32, 45, 50, 43, 72, 45, 78, 67, 72, 93, 62, 117, 78, 107, 108, 93, 142, 83, 148, 117, 130, 155, 108, 195, 120, 173, 162, 147, 208, 117, 222, 163, 192, 213, 158, 267, 158, 243, 212, 205, 270, 163, 300, 205, 258, 267, 212, 333, 192, 317, 258, 267, 328

Offset

3,1

Comments

A trapezoid whose base angles are 60 degrees with larger base b and legs s is denoted by {b X s} here. The regular hexagon is drawn in an equilateral triangular grid and the area of the trapezoid {b X s} is s*(2*b-s) in unit equilateral triangles.

Let the difference between the largest and smallest area of the trapezoids be called the defect. Then a(n) is the minimum defect.

Partitioning of a regular hexagon is done as follows:

Regular hexagon ABCDEF is cut along FC into two identical trapezoids.

Category X (1): Each of the trapezoids is cut parallel to FC into two trapezoids.

Category X (2): Only one of the trapezoids is cut parallel to FC into three trapezoids.

Category Y: Upper trapezoid is cut along PQ parallel to CD in order to form a trapezoid FPQE and a parallelogram PCDQ, where P,Q are points on FC,DE respectively. Finally the parallelogram is cut along MN parallel to BC forming two trapezoids PCNM and MNDQ, where M, N are points on PQ, CD respectively.

Links

Table of n, a(n) for n=3..60.

Janaka Rodrigo, Python Code for Minimum Defects

Example

For n = 4, there are 7 sets of trapezoids:
  {{6 X 1}, {8 X 2}, {8 X 4}, {5 X 1}} with defect = 48-9 = 39,
  {{7 X 1}, {8 X 1}, {6 X 2}, {8 X 4}} with defect = 48-13 = 35,
  {{8 X 1}, {6 X 2}, {8 X 2}, {7 X 3}} with defect = 33-15 = 18,
  {{8 X 1}, {7 X 2}, {8 X 4}, {5 X 1}} with defect = 48-9 = 39,
  {{8 X 1}, {7 X 3}, {8 X 3}, {5 X 1}} with defect = 39-9 = 30,
  {{6 X 2}, {8 X 2}, {8 X 3}, {5 X 1}} with defect = 39-9 = 30,
  {{2 X 1}, {7 X 4}, {8 X 4}, {3 X 1}} with defect = 48-3 = 45.
Therefore a(4) = 18, since this is the minimum defect.
From Hugo Pfoertner, Nov 18 2025: (Start)
a(3) = 13 = 20 - 7:
        *---*---*---*
       /      7      \
      *---*---*---*---*
     /                 \
    *        20         *
   /                     \
  *---*---*---*---*---*---*
   \         11          /
    *---*---*---*---*---*
     \                 /
      *      16       *
       \             /
        *---*---*---*
a(4) = 18 = 33 - 15:
          *---*---*---*---*
         /                 \
        *                   *
       /         33          \
      *                       *
     /                         \
    *---*---*---*---*---*---*---*
   /             15              \
  *---*---*---*---*---*---*---*---*
   \                             /
    *            28             *
     \                         /
      *---*---*---*---*---*---*
       \                     /
        *        20         *
         \                 /
          *---*---*---*---* (End)

Crossrefs

Cf. A390738, A249547.

Sequence in context: A346561 A131019 A380224 * A317771 A037158 A318080

Adjacent sequences: A390736 A390737 A390738 * A390740 A390741 A390742

Keyword

nonn

Author

Janaka Rodrigo, Nov 17 2025

Status

approved