[thijsr]

terrible example,
obviously when the sequence is presented to us by its inventor, there is a hidden assumption that the property of the m^th-rule used to generated the m^th entry (x_m) can't be dependent on m (the structure of the rule that is, it might of course contain m as a variable).
also there is sth of the form (razor or no razor): 'this rule is not ridiculously hard to formulate' in there.

in almost all examples the rule will be rigid and only contain the variables x_{m-1}... x_0 and m.
in more complicated examples the rule itself might change, but only in such a way that that pattern in which it does can be observed in the initial data.

these assumptions are there, b/c it was presented as a puzzle. it's understood as part of what we all agree to be a 'puzzle'.

without the assumptions any random continuation is fine (i guess i should add to the assumption: there is a rule). in a different context (like when you are a scientist looking at nature) you can't rely on such matters and your (well...) point makes some more sense (although i'm sure you're counting on the sun coming up tomorrow). you'll have to find somebody else to debate that though, i've had my share.

anyways... you say they are infinite, i'd say suprise me by getting 1 (alternative one with the assumption that is), no way you can get 2.

(it's obvious your example doesn't work with these assumptions, saying sth to the effect of: use a diffent formula when n>M (and thus referring to M))

cheers,
matt