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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: 92,914

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.**

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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: 92,914

Hi;

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

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

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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: 92,914

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.**

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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: 15,889

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

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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: 92,914

There are formulas of course but he is correct.

**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: 15,889

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,914

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

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

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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: 92,914

Hi;

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

Now on to the next square...

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

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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: 92,914

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...

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

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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: 15,889

He think it does generate the compositions you mentioned.

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,914

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

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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: 92,914

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.

**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: 15,889

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...

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

The knowledge of some things as a function of age is a delta function.

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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: 92,914

Hi;

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

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

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

Hi anonmystefy;

I agree with you.

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