cp's OEIS Frontend

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

%I A214602 #34 Nov 09 2018 20:37:37
%S A214602 9,10,12,14,15,16,18,20,21,22,24,26,27,28,30,32,33,34,35,36,38,39,40,
%T A214602 42,44,45,46,48,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,72,
%U A214602 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
%N A214602 Integer areas of trapezoids such that all sides also have integer lengths.
%C A214602 By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.
%C A214602 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.
%C A214602 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.
%C A214602 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.
%C A214602 The smallest term that is not congruent to 0, 2, 3 or 4 mod 6 (A047229) is 35. - _Alonso del Arte_, Aug 01 2012
%C A214602 _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
%H A214602 Andrew Weimholt, <a href="/A214602/a214602.jpg">Illustration of a trapezoid with parallelograms removed</a>
%e A214602 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.
%Y A214602 Cf. A165513, trapezoidal numbers.
%K A214602 nonn,easy
%O A214602 1,1
%A A214602 _Alonso del Arte_, Jul 22 2012
%E A214602 Missing terms pointed out by _Charles R Greathouse IV_, Aug 02 2012, and _Andrew Weimholt_, Aug 06 2012