[seqfan] Re: Puzzle

Frank Adams-Watters franktaw at netscape.net
Wed Jan 21 03:55:34 CET 2015


Yes, I was mistaken in thinking r must be rational. I was thinking the 
numbers were always in the form (a*r+b)/(c*r+d); but that is not 
correct.

It is true that r must be algebraic.

Franklin T. Adams-Watters

-----Original Message-----
From: hv <hv at crypt.org>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Tue, Jan 20, 2015 8:24 pm
Subject: [seqfan] Re: Puzzle


It is never useful to use the reciprocal twice in a row, nor to start or
end the sequence with it, so we're talking about a sequence [a, b, c, 
... ]
of additions, with a single reciprocal between each set of additions.

When the sequence is length 1 (ie with no reciprocals), the only 
solution
is r = 0; when the sequence is length 2, we have r = 0 or r = (a - b) / 
ab,
with a and b any positive integers.

At length 3, we get an ugly quadratic, which simplifies to the same case
when a + c = b: abcr^2 + (ab - bc)r + (a + c - b) = 0, if I have it 
right.
I guess it'll only get uglier for longer sequences.

The rationals satisfying (a - b) / ab with a, b in Z+ seem like quite
an interesting subset of Q to characterize, though.

I'm not sure "no irrational" is correct though, I think for example
that 1/sqrt(3) works via the sequence [add, rec, add, add, add, rec, 
add].

Hugo

Frank Adams-Watters <franktaw at netscape.net> wrote:
:The best I've got so far is that any a/b where a divides b-1 has this
:property. Add a/b to 1 b*(b-1)/a times, and you get b. Reciprocate to
:1/b, and add a/b another (b-1)/a times; this gets you to 1.
:
:Showing that an r does not have this property is harder. One can show
:that if r does not have the property, neither does r*k for any integer
:k > 0; a step adding r*k can be transformed into k steps adding r. I
:think I can show that 1 does not have the property; it would follow
:that no integer n > 0 has the property. Every number in such a 
sequence
:is a rational function of r (with integer coefficients), so no
:irrational r can have the property.
:
:Franklin T. Adams-Watters
:
:-----Original Message-----
:From: David Wilson <davidwwilson at comcast.net>
:
:Choose any real r >= 0.
:
:Starting with n = 1, on the first step add r, on subsequent steps
:either add
:r or take the reciprocal as you choose.
:
:For example, if r = 1/4, we can generate the sequence
:
:1, 5/4, 3/2, 7/4, 2, 1/2, 3/4, 1.
:
:For which r is it possible to return to 1 as does this sequence?
:
:
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