A390738 - OEIS

%I #11 Nov 23 2025 19:12:21

%S 7,16,28,39,63,75,105,128,156,195,217,272,300,351,399,440,512,544,624,

%T 675,741,820,868,975,1023,1120,1200,1275,1391,1443,1577,1652,1760,

%U 1875,1953,2112,2176,2323,2431,2544,2700,2775,2964,3059,3213,3360,3472,3675,3759,3960,4092,4247,4439,4544,4785,4896

%N 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.

%C 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.

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

%H Janaka Rodrigo, <a href="/A390738/a390738.txt">Python Code for Minimum Defects</a>

%e For n = 5, there are 4 sets of trapezoids

%e   {{10 X 5}, {9 X 4}, {10 X 1}} with defect = 75-19 = 56,

%e   {{10 X 5}, {8 X 3}, {10 X 2}} with defect = 75-36 = 39,

%e   {{10 X 5}, {7 X 2}, {10 X 3}} with defect = 75-51 = 51,

%e   {{10 X 5}, {6 X 1}, {10 X 4}} with defect = 75-11 = 64.

%e Therefore a(5) = 39, since this is the minimum defect.

%Y Cf. A390739.

%K nonn

%O 2,1

%A _Janaka Rodrigo_, Nov 16 2025