Counting n-gons

[Richard Guy]

max area of 1,1,2,2 quad is cyclic, a kite
formed by 2 right triangles with legs 1 & 2,
area 2.      R.

On Tue, 24 Oct 2006, Jonathan Post wrote:

> Dr. George Hockney (ex-Fermilab, now JPL) 
> responded to my suggestion that
> the representative of each equivalence class of 
> deformed polygons with the
> same edge-sequences be that with largest area 
> that: (1)  this might not be
> unique, and (2) that the maximum area (1, 1, 2, 2) 
> quadrilateral is the
> (1+1=2, 2, 2) equilateral triangle of area sqrt 3, 
> where the two edges of
> length 1 make an angle of 180 degrees, and asks if 
> that equilateral triangle
> with a vertex in the middle of its side is really 
> a quadrilateral, or is
> that another ambiguity in equivalence of polygons 
> under geometrical
> similarity? He is also dubious that the 
> integer-edged polygonal knots in R^3
> are really polygons, as I've stated.
>
> I agree with David Wilson's sequence.  What about 
> a sequence table of
> maximal areas of all simple integer-sided k-gons 
> of perimeter n?
>
> -- Jonathan Vos Post
>
> On 10/24/06, David Wilson 
> <davidwwilson at comcast.net> wrote:
>> 
>> And we should have a sequence that counts all 
>> simple integer-sided
>> polygons
>> of perimeter n. Depending on exactly what we are 
>> counting, the sequence
>> might go:
>> 
>> a(3) = 1, the 1-1-1 triangle.
>> a(4) = 1, the 1-1-1-1 quadrilateral.
>> a(5) = 3, the 1-2-2 triangle, 1-1-1-2 
>> quadrilateral, the 1-1-1-1-1
>> pentagon.
>> 
>> etc.
>> 
>> 
>