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From: Robin Becker <robin@reportlab.com>
Subject: Re: A new module for performing tail-call elimination
Date: Thu, 16 Jul 2015 10:28:07 +0100
References: <55a3dcd9$0$3024$426a34cc@news.free.fr> <mailman.532.1436952589.3674.python-list@python.org> <55a76628$0$2846$c3e8da3$76491128@news.astraweb.com>
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Xref: csiph.com comp.lang.python:93918

..........
>
> The point is, people keep insisting that there are a vast number of
> algorithms which are best expressed using recursion and which require TCO to
> be practical, and yet when asked for examples they either can't give any
> examples at all, or they give examples that are not well-suited to
> recursion. Or, at best, examples which are equally good when written either
> using recursion or iteration.
>
> I do believe that such examples surely must exist. But I'm yet to see one.
....
I believe the classic answer is Ackermann's function

http://demonstrations.wolfram.com/RecursionInTheAckermannFunction/

which is said to be not "primitive recursive" ie cannot be unwound into loops; 
not sure whether that implies it has to be recursively defined or can perhaps be 
broken down some other way. For more eye-glazing

http://math.stackexchange.com/questions/75296/what-is-the-difference-between-total-recursive-and-primitive-recursive-functions
-- 
Robin Becker