Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#76 2012-11-05 02:47:15

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Pascal's square

Hi bobbym

The formula I got is (n+2)*2^(n-3) for n>=3. The formula there is (n+3)*2^(n-2). We can see that the difference is only in indexing.


“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

Offline

#77 2012-11-05 02:50:24

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

Hi;

You got that formula how, by curve fitting? That only proves for the values you fit for. It does not mean that formula continues for the next diagonal and the one after 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.

Offline

#78 2012-11-05 03:10:45

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Pascal's square

We can prove by induction that a(i,j)=2^(i+j-2) for i,j>1. From there, it is easy proving the formula...


“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

Offline

#79 2012-11-05 03:15:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

Hi;

What does a(i,j)=2^(i+j-2) for i,j>1 generate?

I would have set it up as


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.

Offline

#80 2012-11-05 03:32:16

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Pascal's square

Sorry, I got 0 starting arrays in my head.


“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

Offline

#81 2012-11-05 03:38:28

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

Hi;

I will leave the inductive proof to you.


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.

Offline

#82 2012-11-05 03:50:35

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Pascal's square

Either way, I think we can be certaing that is how the sequence can be generated...


“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

Offline

#83 2012-11-05 03:59:30

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

Hi;

We can think it all we want. Until we have some proof we ain't gonna convince anybody else of it.


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.

Offline

#84 2012-11-09 21:05:55

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

Re: Pascal's square

Hi everyone;

I tried to find a formula to obtain the sum of the numbers of each diagonal and this is the result:

(2^n)*2+(2^n)2*n, then I simplified it and I obtained 2^(n-1)*n + 2^(n+1). The result is Number of 1's in all compositions of n+1 (A045623 of OEIS), because 2^(n-1)*n + 2^(n+1. Generate the same terms of (n+3)*2^(n-2), the formula of A045623, proposed by anonimnystefy.


Winter is coming.

Offline

#85 2012-11-09 21:21:26

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

Hi;

To refresh my memory, this 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


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.

Offline

#86 2012-11-09 21:50:20

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

Re: Pascal's square

Yes, this one.


Winter is coming.

Offline

#87 2012-11-09 23:37:09

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

Hi;

What is holding up some sort of proof for the problem is that my expression given in post #79 does not cover the first row or the first column.


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.

Offline

#88 2012-11-10 00:00:11

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

Re: Pascal's square

Hi;

So what do you suggest?

Last edited by Mpmath (2012-11-10 00:11:42)


Winter is coming.

Offline

#89 2012-11-10 00:17:01

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

Hi;

I am working feverishly on an expression that actually generates that table. Then it should be easier to prove the relation is true.


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.

Offline

#90 2012-11-10 00:45:59

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

Re: Pascal's square

Hi;

Ok. Thanks.


Winter is coming.

Offline

#91 2012-11-10 10:38:24

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

Hi;

Nothing yet.


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.

Offline

#92 2012-11-10 11:10:57

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

Re: Pascal's square

Hi;

Ok. Keep me informed, thanks.

Last edited by Mpmath (2012-11-10 11:11:39)


Winter is coming.

Offline

#93 2012-11-10 13:17:22

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

I finally got an expression that will generate the table but it is too complicated for me to understand. At least we can see more of the table.


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.

Offline

#94 2012-11-10 22:20:35

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

Re: Pascal's square

Hi bobbym;

Good job! Can I see the expression?


Winter is coming.

Offline

#95 2012-11-10 22:34:03

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

Hi;

I did not post it because it is virtually useless.


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.

Offline

#96 2012-11-10 22:48:12

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Pascal's square

That is equal to 2^(i-2)*2^(j-2).


“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

Offline

#97 2012-11-10 22:51:54

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

Aha! You went for the trap. It is incorrect for

It is also incorrect for the whole first column and first row.


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.

Offline

#98 2012-11-10 22:57:01

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

Re: Pascal's square

Hi;

Well, good job!


Winter is coming.

Offline

#99 2012-11-10 23:02:46

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Pascal's square

bobbym wrote:

Aha! You went for the trap. It is incorrect for

It is also incorrect for the whole first column and first row.

I still think we could use 2^(i-1) and 2^(j-1) for the first column and row respectively, and 2^(i-1) for the rest.


“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

Offline

#100 2012-11-10 23:20:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,347

Re: Pascal's square

The only problem is that Mathematica gagged on both those series.

I want to sum along the diagonals but if it takes two functions for every diagonal that is going to make the proof much harder or impossible.


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.

Offline

Board footer

Powered by FluxBB