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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

That is very good, you kept the dominant terms.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

Exactly. They seem to do the job good enough. I will try getting the closed form analitically today.

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Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

Uh wait. That is the analytical answer up there. So you will looking for a shorter one?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

You didn't get it by analityc methods. You used M's FindSequenceFunction command. I want to see if I can get the answer analitically.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

All you provided me was with your original algorithm. I had to use the sequence. And why not just try induction and prove the a_n is correct?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

Do you think there is a nicer reccurence for the sequence?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

Maybe. The weirdest thing is that sequences can have more than one difference equation.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

I have the difference equation. It's a[n+4]-4a[n+2]-a[n]+4a[n-2]=0. Don't ask how I got it!

*Last edited by anonimnystefy (2013-04-17 04:41:07)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

I am going to ask just that.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

It is a bit tough to explain. I played a little bit of spot-the-pattern...

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

The solution to that will probably be longer than the other one.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

I think you will get the same solution.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

Possibly, or the trigonometric terms will be replaced by (-1)^n or something like that.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

The trigonometric terms will be replaced by i^n and (-i)^n.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

Yes, that is possible too but I would not exactly call that simpler.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

True. But the reduced form looks really nice, though!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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It is an asymptotic form yes? It is supposed to be shorter and easier to compute.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

Yes. It looks much much prettier than the nasty exact one.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

But you possibly missed the forest for the trees.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

How?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

It appears there might be a pattern worth investigating that is much simpler. It may not hold but it is worth a look. You have to get out of the box axiomatic math puts you in. It is small, dark and has little air.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

Which pattern? In the binary representation? That is the one I used to get the difference equation.

Hm, could you do asymptotic_a[n]-a[n] for me?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

One question at a time can be investigated well. This one is more important.

Which pattern? In the binary representation?

No, not that one. Look below, is there anything cooking?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

Well, they approach 1. That is why my formula is an asymptotic one. I also notice that every fourth term repeats two terms later. I do not notice anything else.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

Yes, the differences between the asymptotic form and the actual answer follow a pattern.

1,3, then 1,1,1,3...

This suggests 2 recurrences and just adding the appropriate constant. Now you would have a short exact answer.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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