[seqfan] Re: Fwd: Re: A000207: Maple-Program wrong

[Rainer Rosenthal]

Maximilian Hasler wrote:

> (Do you really mean A207 ?) -- (*) this may be your problem

Oh no, it's really A000207, I am interested in. When I said
the Maple program was wrong, I simply meant that it didn't work
when simply cut and pasted.

> A000207 := proc(n) local k, it1, it2;
>  if n=1 then 1 else
>  k := floor(n/2)+1;
>  if n mod 2 = 0 then it1 := A000108(n/2+1-2)/4 else it1 := 0 fi:
>  if n mod 3 = 0 then it2 := A000108(n/3+1-2)/3 else it2 := 0 fi:
>  A000108(n-2)/(2*n) + it1 + A000108(k-2)/2 +  it2
>  fi
> end:

This one works perfectly and agrees with the sequence.
Many thanks to you, Richard and Peter for your corrections. I'd be
glad if you were to submit the small version above or the even more
packed version from Peter Luschny.

***

As I said I've come across A000207 when I counted these figures, made
up of n copies of an equilateral triangle:

         .                . - .               . - .
        / \              / \ /               / \ / \
       . - .            . - .               . - . - .
_____________________________________________________________
    n=1 only 1          n=2 only 1          n=3 only 1
      figure              figure              figure


         .                                  . - .
        / \                                / \ / \
       . - .            . - . - .         . - . - .
      / \ / \          / \ / \ /               \ /
     . - . - .        . - . - .                 .
______________________________________________________________
            n=4 with 3 different figures

Thanks to the OEIS and thanks to the working Maple programs I was able
to see how the number of figures is growing.  I am quite astonished
to find the rate of growth approximating the value 4.
For large n there are about 4 times more polynomioes with n+1 triangles
than with n triangles, or differently put: large polynomioes have an
average of 4 extensions.

Do you think it would be an interesting comment that

            lim{n->oo} A000207(n+1)/A000207(n) = 4

or is this an immediate consequence for the learned reader of the fact
that the g.f. produces real values for -1/2 <= x <= 1/4?

Best regards,
Rainer