A214602 Integer areas of trapezoids such that all sides also have integer lengths.

9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106

Offset: 1

Author

Alonso del Arte, Jul 22 2012

Keywords

• nonn
• easy

Comments

By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.

If from an isosceles trapezoid having all sides with integer lengths we remove the widest rectangle having the same height as the trapezoid, we are left with two triangles that both correspond to the same Pythagorean triple.

Another possibility if we can remove a rectangle with the same width as the top of the trapezoid is that the remaining two triangles will correspond to two different Pythagorean triples both having the same smallest term, e.g., (15, 20, 25) and (15, 30, 36); this trapezoid has a base 51 units long, a top 1 unit long, height 15 units, left side 25 units and right side 36 units.

The smallest term that corresponds to more than one trapezoid is 15, which can be the area of a right trapezoid with a base measuring 7 units, a top of 3 units, height and left (or right) side 3 units, and right (or left) side 5 units; or an isosceles trapezoid with a base 9 units, top 1 unit, height 3 units, and left and right sides 5 units each.

The smallest term that is not congruent to 0, 2, 3 or 4 mod 6 (A047229) is 35. - Alonso del Arte, Aug 01 2012

Andrew Weimholt has pointed out that it is also possible to construct a trapezoid with the requirements above from which a rectangle can't be removed to leave two right triangles: one way to do this is to join two triangles corresponding to two different Pythagorean triples and then remove a parallelogram with two sides each measuring one less than the smallest number in the smaller Pythagorean triple. See Weimholt's illustration. - Alonso del Arte, Aug 06 2012

Examples

			21 is in the sequence because it is the area of a trapezoid with a base measuring 11 units, a top of 3 units, and left and right sides of 5 units each.
		

Links

• Andrew Weimholt, Illustration of a trapezoid with parallelograms removed

Crossrefs

Cf. A165513, trapezoidal numbers.

Extensions

Missing terms pointed out by Charles R Greathouse IV, Aug 02 2012, and Andrew Weimholt, Aug 06 2012