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#1 2012-12-25 05:12:32

bobbym
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A nice diophantine equation.

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 have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#2 2012-12-26 07:49:31

anonimnystefy
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Re: A nice diophantine equation.

It is also in the OEIS.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#3 2012-12-26 07:52:14

bobbym
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Re: A nice diophantine equation.

Hmmmm. What did you think I meant?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#4 2012-12-26 08:00:38

anonimnystefy
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Re: A nice diophantine equation.

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


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#5 2012-12-26 08:03:13

bobbym
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Re: A nice diophantine equation.

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 have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#6 2012-12-26 08:05:41

anonimnystefy
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Re: A nice diophantine equation.

I see that it is from Norse mythology...


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#7 2012-12-26 19:33:43

bobbym
Administrator

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Re: A nice diophantine equation.

Yes, it is similar to the Oracle at Delphi.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#8 2012-12-26 20:11:18

anonimnystefy
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Re: A nice diophantine equation.

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


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#9 2012-12-26 20:15:00

bobbym
Administrator

Online

Re: A nice diophantine equation.

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


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#10 2012-12-26 20:21:02

anonimnystefy
Real Member

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Re: A nice diophantine equation.

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


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#11 2012-12-26 20:23:37

bobbym
Administrator

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Re: A nice diophantine equation.

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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#12 2012-12-26 20:28:03

anonimnystefy
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Re: A nice diophantine equation.

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


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#13 2012-12-26 20:28:42

bobbym
Administrator

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Re: A nice diophantine equation.

What did you think of the problem?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#14 2012-12-26 20:30:34

anonimnystefy
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Re: A nice diophantine equation.

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.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#15 2012-12-26 20:34:11

bobbym
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Re: A nice diophantine equation.

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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#16 2012-12-26 20:39:18

anonimnystefy
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Re: A nice diophantine equation.

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


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#17 2012-12-26 20:41:11

bobbym
Administrator

Online

Re: A nice diophantine equation.

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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#18 2012-12-26 20:47:21

anonimnystefy
Real Member

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Re: A nice diophantine equation.

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


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#19 2012-12-26 20:51:06

bobbym
Administrator

Online

Re: A nice diophantine equation.

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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#20 2012-12-26 20:52:08

anonimnystefy
Real Member

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Re: A nice diophantine equation.

Yes, I remember writing it down, but have no idea where...

Last edited by anonimnystefy (2012-12-26 20:56:05)


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#21 2012-12-26 20:53:12

bobbym
Administrator

Online

Re: A nice diophantine equation.

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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#22 2012-12-26 20:56:35

anonimnystefy
Real Member

Offline

Re: A nice diophantine equation.

No need. It can be found on Wikipedia.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#23 2012-12-26 21:52:36

bobbym
Administrator

Online

Re: A nice diophantine equation.

And there are other ways to do this problem.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#24 2012-12-26 21:54:09

anonimnystefy
Real Member

Offline

Re: A nice diophantine equation.

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


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#25 2012-12-26 21:55:55

bobbym
Administrator

Online

Re: A nice diophantine equation.

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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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