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J. J. Foncannon
science forum beginner

Joined: 10 Jun 2005
Posts: 3

Posted: Fri Jun 10, 2005 3:19 pm    Post subject: very annoying sequence!!!!

A sequence s[n] is defined as follows:

s[1]=1, s[2]=12, s[3]=20,

ans

s[n+3]=2s[n+2] + 2s[n+1] - s[n], n=1, 2, ... .

PROVE: 1 + 4s[n] s[n+1] is a perfect square.

--

__________________________________________________
**********************************************************
J. J. Foncannon

The Belgian surrealist painter Renee Magritte entered a cheese store in
Brussels to purchase a wheel of Swiss cheese. The owner pulled a wheel
from the front window, but Magritte said he preferred the one on the
back counter.
“But they are identical,” the owner protested.
“No,” Magritte insisted. “This one’s been stared at.”
**********************************************************
Timothy Little
science forum Guru Wannabe

Joined: 30 May 2005
Posts: 295

Posted: Sat Jun 11, 2005 10:01 am    Post subject: Re: very annoying sequence!!!!

J. J. Foncannon wrote:
 Quote: A sequence s[n] is defined as follows: s[1]=1, s[2]=12, s[3]=20, s[n+3]=2s[n+2] + 2s[n+1] - s[n], n=1, 2, ... . PROVE: 1 + 4s[n] s[n+1] is a perfect square.

Can you prove that
1 + 4 s[n+1] s[n+2] = ((s[n+3] - s[n]) / 2)^2
for n >= 1? For example, by induction?

- Tim
J. J. Foncannon
science forum beginner

Joined: 10 Jun 2005
Posts: 3

Posted: Sat Jun 11, 2005 5:56 pm    Post subject: Re: very annoying sequence!!!!

It works.... How did you know to try this?
Thanks,
Jet

Timothy Little wrote:

 Quote: J. J. Foncannon wrote: A sequence s[n] is defined as follows: s[1]=1, s[2]=12, s[3]=20, s[n+3]=2s[n+2] + 2s[n+1] - s[n], n=1, 2, ... . PROVE: 1 + 4s[n] s[n+1] is a perfect square. Can you prove that 1 + 4 s[n+1] s[n+2] = ((s[n+3] - s[n]) / 2)^2 for n >= 1? For example, by induction? - Tim

--

__________________________________________________
**********************************************************
Jet Foncannon

The Belgian surrealist painter Renee Magritte entered a cheese store in
Brussels to purchase a wheel of Swiss cheese. The owner pulled a wheel
from the front window, but Magritte said he preferred the one on the
back counter.
“But they are identical,” the owner protested.
“No,” Magritte insisted. “This one’s been stared at.”
**********************************************************
J. J. Foncannon
science forum beginner

Joined: 10 Jun 2005
Posts: 3

Posted: Sun Jun 12, 2005 12:14 am    Post subject: Re: very annoying sequence!!!!

Very good work, Tim. The induction needed to verify your assertion is
quite straightforward, and I won't bore you with it.
Another fact: the square root of 1+4s[n]s[n+1] satisfies the same
recurrence as s[n].
Jet

Timothy Little wrote:

 Quote: J. J. Foncannon wrote: It works.... How did you know to try this?

Jet

 Quote: First I worked out 10 or so terms of the series and verified that they were in fact perfect squares. So next I looked at the square roots. Then I took a detour into working out the closed form for the s[n], which I found, but couldn't see anything in it that obviously led to your expression being a perfect square. Dead end so far (though more work might have succeeded). So, back to eyeballing the list of square roots again. I expected that their sequence would be some linear combination of some s[n]'s, because 1) they were roughly the square root of a product of two terms, and so should be roughly linear in the size of the sequence terms, and 2) the recurrence for the sequence was linear. With that in mind I did notice that they were usually pretty close to half of s[n+3], but a bit smaller. So I looked at (s[n+3] - something) / 2, and the 'something' for the first few happened to be s[n]. If I had been looking at s[n+2] instead, I would likely have eventually ended up with the equivalent expression (s[n+2] + s[n+1] - s[n]). There is no doubt a less haphazard way to get the same result, but I didn't find it :) - Tim

--

__________________________________________________
**********************************************************
J. J. Foncannon

The Belgian surrealist painter Renee Magritte entered a cheese store in
Brussels to purchase a wheel of Swiss cheese. The owner pulled a wheel
from the front window, but Magritte said he preferred the one on the back
counter.
“But they are identical,” the owner protested.
“No,” Magritte insisted. “This one’s been stared at.”
**********************************************************
Timothy Little
science forum Guru Wannabe

Joined: 30 May 2005
Posts: 295

Posted: Sun Jun 12, 2005 12:23 am    Post subject: Re: very annoying sequence!!!!

J. J. Foncannon wrote:
 Quote: It works.... How did you know to try this?

First I worked out 10 or so terms of the series and verified that they
were in fact perfect squares. So next I looked at the square roots.

Then I took a detour into working out the closed form for the s[n],
which I found, but couldn't see anything in it that obviously led to
your expression being a perfect square. Dead end so far (though more
work might have succeeded).

So, back to eyeballing the list of square roots again. I expected
that their sequence would be some linear combination of some s[n]'s,
because 1) they were roughly the square root of a product of two
terms, and so should be roughly linear in the size of the sequence
terms, and 2) the recurrence for the sequence was linear.

With that in mind I did notice that they were usually pretty close to
half of s[n+3], but a bit smaller. So I looked at
(s[n+3] - something) / 2, and the 'something' for the first few
happened to be s[n]. If I had been looking at s[n+2] instead, I would
likely have eventually ended up with the equivalent expression
(s[n+2] + s[n+1] - s[n]).

There is no doubt a less haphazard way to get the same result, but I
didn't find it :)

- Tim

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