A390738 a(n) is the least possible difference between the areas of the largest and smallest trapezoids in a 3 element set of distinct integer-sided trapezoids whose base angles are 60 degrees that fill a regular hexagon of side n units.

7, 16, 28, 39, 63, 75, 105, 128, 156, 195, 217, 272, 300, 351, 399, 440, 512, 544, 624, 675, 741, 820, 868, 975, 1023, 1120, 1200, 1275, 1391, 1443, 1577, 1652, 1760, 1875, 1953, 2112, 2176, 2323, 2431, 2544, 2700, 2775, 2964, 3059, 3213, 3360, 3472, 3675, 3759, 3960, 4092, 4247, 4439, 4544, 4785, 4896

Offset

2,1

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.

Links

Table of n, a(n) for n=2..57.

Janaka Rodrigo, Python Code for Minimum Defects

Example

For n = 5, there are 4 sets of trapezoids
  {{10 X 5}, {9 X 4}, {10 X 1}} with defect = 75-19 = 56,
  {{10 X 5}, {8 X 3}, {10 X 2}} with defect = 75-36 = 39,
  {{10 X 5}, {7 X 2}, {10 X 3}} with defect = 75-51 = 51,
  {{10 X 5}, {6 X 1}, {10 X 4}} with defect = 75-11 = 64.
Therefore a(5) = 39, since this is the minimum defect.

Crossrefs

Cf. A390739.

Sequence in context: A326664 A028560 A190530 * A345071 A351044 A133694

Adjacent sequences: A390735 A390736 A390737 * A390739 A390740 A390741

Keyword

nonn

Author

Janaka Rodrigo, Nov 16 2025

Status

approved