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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

Hi everyone;

I have carefully studied all Pascal's squares and in some of them I found some interesting successions. In this square, that MathIsFun proposed:

1 1 1 1

1 3 5 7

1 5 13 25

1 7 25 63

The sum of the numbers of every diagonal give Pell numbers (A000129 of OEIS).

1=1

1+1=2

1+3+1=5

1+5+5+1=12

1+7+13+7+1=29

Regarding my Pascal's square:

1 1 2 4 8

1 1 2 4 8

2 2 4 8 16

4 4 8 16 32

8 8 16 32 64

16 16 32 64 128

The sum of the numbers of every diagonal give numbers of 1's in all compositons of n+1 (A045623 of OEIS).

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,267

Hi Mpmath;

For the Pell number square did you try generating a bigger square and checking that?

**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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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

Hi;

Yes, I try. Numbers coincide. I notice right now that MathIsFun square is square array of Delannoy numbers (A008288 of OEIS).

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,267

Hi;

Okay, I know the Delannoy numbers. Thanks for spotting that.

**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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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

You're welcome.

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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

At least we find an alternative system to obtain square array of Delannoy numbers!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,267

Yes, I am looking at that right now. It could be of some importance.

**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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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

Ok. Let me know if you find something interesting.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Well, isn't it the same way Delannoy numbers are made, only with graphical representation?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,267

There are formulas of course but he is correct.

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: 15,544

Who is correct about what?

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: 87,267

You are correct in that is how they are generated in a lattice. So therefore of course that works by definition.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

Hi bobbym;

So what are your conclusions?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,267

Hi;

Not a whole lot, but that MIF square does generate Delannoy numbers.

Now on to the next square...

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

Yes...?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,267

Hi;

1 1 2 4 8

1 1 2 4 8

2 2 4 8 16

4 4 8 16 32

8 8 16 32 64

16 16 32 64 128The sum of the numbers of every diagonal give numbers of 1's in all compositons of n+1.

I see A152195 and A177992 but they are not the same as your square.

Does seem to be generating the compositions. That is interesting...

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

Hi;

Sorry, but I don't understand what you mean when you say "Does seem to be generating the compositions.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

He think it does generate the compositions you mentioned.

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: 87,267

Hi;

As far as I have gone it does. That does not mean the pattern will continue. To show that it does I need a mathematical proof. So, all I can say right now is that it seems to. I can not be more definite than that.

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

Hi;

Now I understand. Thanks!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,267

Hi;

Still, I like the way you look for patterns in numbers. This type of empirical research although not rigorous enough for mathematical proof is the way discoveries are often made.

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: 15,544

bobbym wrote:

Hi;

As far as I have gone it does. That does not mean the pattern will continue. To show that it does I need a mathematical proof. So, all I can say right now is that it seems to. I can not be more definite than that.

In oeis, there is a formula with tthat sequence that matches with a formula for the diagonal sum of the square, so I would say that that is proof enough...

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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

Hi bobbym;

Thanks for compliment.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,267

Hi;

Matches? In what way? You have a formula for the diagonal sum of that square?

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

Combinatorics is Algebra and Algebra is Combinatorics.

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**Mpmath****Member**- Registered: 2012-10-11
- Posts: 216

Hi anonmystefy;

I agree with you.

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