[seqfan] Re: Trapezoidal area integers: overlooked types

[Alonso Del Arte]

Then yes, squares, rectangles and parallelograms have to be excluded or
this just becomes a pointless duplicate of A27. I have added a remark to
that effect.

If the (6, 8, 10), (8, 15, 17) with inserted 8L rectangle is a minimal case
for that kind, then I'm OK: 84, 92, 100, 108, ... are already in A214602
for other kinds of trapezoids. But the remark "The smallest term that does
not correspond to a right or isosceles trapezoid..." will have to be
revised.

As for the trapezoids you mention in the last paragraph, I'm going to have
to draw them on paper; I'm sure you're correct but I can't quite visualize
in my mind at the moment.

Al

On Mon, Aug 6, 2012 at 4:04 AM, Andrew Weimholt
<andrew.weimholt at gmail.com>wrote:

> On Sat, Aug 4, 2012 at 1:33 PM, Alonso Del Arte
> <alonso.delarte at gmail.com> wrote:
> >
> > Have I overlooked any other kinds of trapezoids than can have integers
> for
> > area and all four sides? And if so, does this mean that almost all
> > composite numbers are in A214602?
> >
>
> If "trapezoid" is defined as a quadrilateral with "at least one" pair of
> parallel edges, then squares, rectangles, and parallelograms are also
> trapezoids, in which case every positive integer should be included.
> (a 1 by L rectangle has integer sides: 1,L,1,L, and integer area L)
>
> However, if we define "trapezoid" as a quadrilateral with "exactly one"
> pair of parallel edges, then there are still some that you missed...
>
> It is not true that after removing the largest rectangle, the two
> remaining pythagorean triangles must have the same length for their
> smallest side. Take the pythagorean triangles (6,8,10) and (8,15,17).
> We can put an 8 x L rectangle in between them, and we get a rectangle
> with area 84+8L.
>
> It is also a faulty assumption that the non parallel sides must have
> slopes with opposite signs with respect to the base. We can attach
> the (6,8,10) and (8,15,17) directly to each other with no rectangle in
> between, and still get a trapezoid (with area 84) by flipping one of
> the triangles vertically (sides are 6,10,15,17).  Furthermore, we
> can remove anywhere from 1 to 5 units of length from the parallel
> edges from this trapezoid and still get a trapezoid with unit area.
> We are merely removing parallelograms with unit area instead of
> rectangles. If we remove 5 units, we get a trapezoid with area 44,
> and sides (1,10,10,17)
>
> Andrew
>
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Alonso del Arte
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