a graph puzzle

[Max Alekseyev]

Well, Maximilian seemed to be the only person got interested in this puzzle.

I think, now it is time to explain what it is.

The graph represents a mapping between integer pairs of co-prime
integers: (p,q) -> (a,b), where (a,b) is the ordered pair of the
absolute values of "smallest" Bezout coefficients for the pair (p,q).
To find (a,b), one need to find Bezout identity:
u*p + v*q = 1
with the minimum value of |u|+|v|, and take the ordered pair of values
|u|,|v| (the smallest first) as (a,b).

Alternatively, (a,b) can be computed as the numerator and denominator
of the second but last convergent to the fraction p/q.
Sample implementation in PARI/GP is below:

{ ab(p,q) = contfracpnqn(contfrac(p/q))[,2]~ }

? ab(23,30)
%1 = [10, 13]

So, my question remains the same - is this mapping present in OEIS in
either form?

P.S. My original concern was about pairs below given N, having the
longest path to (0,1), but they seem to be simply consecutive
Fibonacci numbers (as in Lame's theorem about Euclidean algorithm).

Regards,
Max

On Sat, Mar 22, 2008 at 6:10 AM, Max Alekseyev <maxale at gmail.com> wrote:
> Good morning,
>
>  I've generated a graph on the unordered co-prime pairs of non-negative
>  integers below 30,
>  using some quite "natural" mapping.
>
>  The graph has quite nice and somewhat fractal structure:
>  http://img505.imageshack.us/img505/4065/3cbv7.jpg  (its size is ~ 800KB)
>
>  Can you guess what is the underlying mapping "hidden" in this graph?
>
>  Regards,
>  Max
>
>  P.S. I hate puzzles like this one but maybe you like ;)
>
>  P.S. Actually, my concern is whether this mapping is present in OEIS
>  in either form.
>