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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Hi and Happy Holiday;

This equation appeared on another forum awhile back.

with a,b,c >= 0 and a,b,c are integers. The OP wanted a closed form for that. One that depends on n. They were unable to answer the question, let's see what experimental mathematics can do.

The first thing we notice is that the generating function for this sequence is

Using the methods in another thread

http://www.mathisfunforum.com/viewtopic.php?id=16137

we can come up with an asymptotic form of

This is actually quite good. So at least we have an approximate answer if we can not get a closed form as the OP requests. Having that in hand we go after the big fish, "The Closed Form!"

We start by using the fact that programming can often solve problems when math can not. We write a little proggy to brute force the solutions to

for n=1,2,3...

we get this sequence of numbers:

{0,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,8,9,9,11,11,12,13,14,15,16,17,18,19,21,21,23,24,25,27,28,29,31,32,34,35,37,38,40,42,43,45,47,48,51,52,54,56,58,60,62,64,66,68,71...}

We go over to Mimir's Fountain and if you have been following these threads you know where that is too and ask it about this sequence. The good thing is we are not required to part with an eye for the information.

One of the advantages of being able to program in some language is the ability to query the well for its wisdom.

It tells us that the closed form is:

Not a bad result and a good answer for the OP. Maybe someday she will learn about this forum...

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

It is also in the OEIS.

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: 88,714

Hmmmm. What did you think I meant?

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

I didn't know what Mimir's fountain was...

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: 88,714

Okay, you did not understand the reference. I got it from the same place.

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

I see that it is from Norse mythology...

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: 88,714

Yes, it is similar to the Oracle at Delphi.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

I thought Java was associated with Oracle, not Delphi...

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Nope, this was way before Java. Way before me even.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

I remember reading that Eclipse, the Java IDE, was produced by Oracle (now Sun). I never knew they also produced a Delphi IDE.

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Might be true but the point is the two Delphi's have nothing in common.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

Ahhhh... Well they have one thing in common - their names!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

What did you think of the problem?

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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It is interesting. Looking for the closed form just didn't feel right, though... I like your approximation better. It's easier to remember and to get.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

The way I used is a little clumsy but asymptotic analysis is a couple of levels above me. I

use my own idea there and it does work. gAr and I were playing around with a formula that was

more to the point.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

Which formula is that? Is it the one with the gamma function and some other stuff?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Yes, that is the one. The example is for the catalan numbers.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

I also saw the example of n! and I think gAr used it on the 2n choose n sequence.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

May have, it only works on a particular form though. The above method works on more.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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Yes, I remember writing it down, but have no idea where...

*Last edited by anonimnystefy (2012-12-25 21:56:05)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Hmmmm. You and Agnishom seem similar in that respect. I can post it.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
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No need. It can be found on Wikipedia.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

And there are other ways to do this problem.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,603

The formula gives n^2/180 as the approximation...

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,714

Now you know formulas mean nothing to me. Theorems even less. What counts is practicality, usefulness.

Does that work?

**In mathematics, you don't understand things. You just get used to them.**

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