A390523 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 an equilateral triangular grid of side n units.

6, 5, 9, 9, 8, 7, 9, 8, 12, 16, 12, 16, 15, 10, 20, 24, 24, 6, 19, 26, 31, 20, 14, 19, 32, 35, 28, 24, 20, 28, 32, 40, 21, 41, 43, 22, 45, 41, 39, 24, 29, 47, 60, 50, 23, 36, 45, 58, 48, 47, 21, 59, 65, 64, 36, 49, 64, 45, 72, 53, 51, 54, 40, 69, 83, 76, 45, 55, 48, 84, 85

Offset

5,1

Comments

Let the difference between the largest and smallest areas of the trapezoids be called the defect. Then a(n) is the minimum defect. The area of a trapezoid {b X s} is s*(2*b-s).

Links

Table of n, a(n) for n=5..75.

Example

For n = 6, there are 9 sets of trapezoids:
  {{2 X 1}, {3 X 1}, {3 X 2}, {6 X 2}} with defect = 20-3 = 17,
  {{2 X 1}, {4 X 1}, {6 X 1}, {4 X 3}} with defect = 15-3 = 12,
  {{3 X 1}, {3 X 2}, {6 X 1}, {4 X 2}} with defect = 12-5 = 7,
  {{2 X 1}, {3 X 1}, {4 X 2}, {5 X 2}} with defect = 16-3 = 13,
  {{2 X 1}, {3 X 1}, {4 X 1}, {5 X 3}} with defect = 21-3 = 18,
  {{2 X 1}, {3 X 2}, {5 X 1}, {5 X 2}} with defect = 16-3 = 13,
  {{3 X 1}, {4 X 1}, {3 X 2}, {5 X 2}} with defect = 16-5 = 11,
  {{3 X 1}, {4 X 1}, {5 X 1}, {4 X 3}} with defect = 15-5 = 10,
  {{4 X 1}, {3 X 2}, {5 X 1}, {4 X 2}} with defect = 12-7 = 5.
Therefore a(6) = 5, since this is the minimum defect.

Crossrefs

Cf. A389392, A389518.

Sequence in context: A073230 A134881 A229983 * A394985 A251859 A100884

Adjacent sequences: A390520 A390521 A390522 * A390524 A390525 A390526

Keyword

nonn

Author

Janaka Rodrigo, Nov 09 2025

Status

approved