[seqfan] Re: New index-entry for fraction trees.

[Kevin Ryde]

antti.karttunen at gmail.com (Antti Karttunen) writes:
>
> Not knowing better, I tentatively named some of these with the surnames of
> their submitters,
> e.g. ... Wilson ...

That one, if I have the right one, also

  Shen Yu-Ting, "A Natural Enumeration of Non-Negative Rational Numbers
  -- An Informal Discussion", American Mathematical Monthly, 87, 1980,
  pages 25-29.  http://www.jstor.org/stable/2320374

  D. N. Andreev, "On a Wonderful Numbering of Positive Rational Numbers",
  Matematicheskoe Prosveshchenie, Ser. 3, 1, 1997, pages 126-134.
  http://mi.mathnet.ru/mp12

> ... or fraction tree systems still missing from that list,

Chan A191379 makes a 3-ary tree (or forest, since 2 roots), but only X/Y
with X,Y not both odd, or something like that, and not necessily in
least terms.  It generalizes Calkin-Wilf so one sequence a(n)/a(n+1).
There's 4-ary, 5-ary, etc forms too but no entries for them I know.
I think even-ary gives all rationals, still not necessarily in least
terms.  The trees of primitive pythagorean triples are X/Y not both odd
if such subsets are of interest.  I imagine indexed elsewhere already.

> (What does HCS mean, by the way?)

Oh that would be from me circuitously, in absense of knowing better (and
still not knowing better :).  It's supposed to be Paul D. Hanna in
A071766 here circa 2002, plus

  Jerzy Czyz and William Self, "The Rationals Are Countable: Euclid's
  Proof", The College Mathematics Journal, volume 34, number 5,
  November 2003, page 367.
  http://www.jstor.org/stable/3595818

I expect there's room to list the names in the index, to help whichever
someone might know it from.  The abbreviation was supposed to be
shortish for program code, then in my document.

For interest, a bit of exposition on this tree at

  http://www.cut-the-knot.org/do_you_know/countRatsCF.shtml

and there was also, but maybe slightly down or gone

  http://www.dm.unito.it/~cerruti/doc-html/tremattine/tre_mattine.pdf
unless at
  http://www.academia.edu/5233434/Numeri