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## #51 2012-11-03 09:26:14

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

### Re: Pascal's square

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).

Winter is coming.

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## #52 2012-11-03 09:42:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Pascal's square

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #53 2012-11-03 09:50:57

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

### Re: Pascal's square

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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## #54 2012-11-03 09:57:54

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Pascal's square

Hi;

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

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #55 2012-11-03 10:01:07

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

### Re: Pascal's square

You're welcome.

Winter is coming.

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## #56 2012-11-03 10:10:07

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

### Re: Pascal's square

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

Winter is coming.

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## #57 2012-11-03 10:21:47

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Pascal's square

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #58 2012-11-03 10:58:45

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

### Re: Pascal's square

Ok. Let me know if you find something interesting.

Winter is coming.

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## #59 2012-11-03 11:33:37

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Pascal's square

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

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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## #60 2012-11-03 11:35:52

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Pascal's square

There are formulas of course but he is correct.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #61 2012-11-03 11:54:15

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Pascal's square

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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## #62 2012-11-03 11:56:02

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Pascal's square

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #63 2012-11-04 09:52:01

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

### Re: Pascal's square

Hi bobbym;

Winter is coming.

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## #64 2012-11-04 10:33:25

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Pascal's square

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #65 2012-11-04 21:07:27

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

### Re: Pascal's square

Yes...?

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## #66 2012-11-05 01:31:56

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Pascal's square

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  128

The 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #67 2012-11-05 02:16:28

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

### Re: Pascal's square

Hi;

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

Winter is coming.

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## #68 2012-11-05 02:19:30

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Pascal's square

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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## #69 2012-11-05 02:22:11

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Pascal's square

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #70 2012-11-05 02:24:53

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

### Re: Pascal's square

Hi;

Now I understand. Thanks!

Winter is coming.

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## #71 2012-11-05 02:30:50

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Pascal's square

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #72 2012-11-05 02:35:07

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Pascal's square

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...

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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## #73 2012-11-05 02:38:20

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

### Re: Pascal's square

Hi bobbym;

Thanks for compliment.

Winter is coming.

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## #74 2012-11-05 02:39:24

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Pascal's square

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #75 2012-11-05 02:40:23

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

### Re: Pascal's square

Hi anonmystefy;

I agree with you.

Winter is coming.

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