Pascal's Triangle
 

Coefficients in the expansion of (x+y)^n form a pattern which is useful for small values of n. It is known as Pascal's triangle.

n Pascal's triangle
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1

Write a Matlab script to calculate and display this expression in the form of the triangle for the first 10 values of n. Use the sprintf command to display in a suitable format such as that shown above.





% A Program for Pascal's Triangle

if nargin ~=2
n=1:10;
s=4;
end

if n<55
a= n/10;
x=0;
y=0;
x2=0;
w=1;


hold on;

for m = 1:n-1
x = x2-a/2;
y = y-a/2*sqrt(2);
w = 1;

x2 = x;

for l = 1:m
x = x+a;
w = nchoosek(m,1);

end
end

else clf;
text(-0.2,0.5,' FontSize',23);
end;

hold off;
axis square;
axis equal;
axis off;


I had tried the question but i am not getting through it can any1 help me out in this problem.

Re: Pascal's Triangle
 

Try to plan out a flow chart first.

Re: Pascal's Triangle
 

hello i want to write code for this
in my opinion pascal triangle is:-
1
1 1
1 2 1
1 3 3 1

u wil find this by following prog:-
/*Program to Calculate the Pascal triangle */
#include<stdio.h>
int main()
{
int a[20][20];
int i,j,c,n,k;
printf("Enter how many lines do you want");
scanf("%d",&n);
a[1][1]=1;
a[2][1]=1;a[2][2]=1;
for(i=1;i<=n;i++)
{ for(j=1;j<=n-i;j++)
printf(" ");
if(i<3)
{
for(k=1;k<=i;k++)
{
printf("%d ",a[i][k]);
}
printf("\n");
}
else
{
a[i][1]=1;
printf("%d ",a[i][1]);
j=2;c=3;
while(j<i)
{
a[i][j]=a[i-1][c-2]+a[i-1][c-1];
printf("%d ",a[i][j]);
c=c+1;
j=j+1;
}
a[i][j]=1;
printf("%d\n ",a[i][j]);
}
}
getche();
}

Re: Pascal's Triangle
 

These seem rather elaborate in comparison to something like (in J):
Pascal =: i. !/ i.

See http://www.jsoftware.com/jwiki/NYCJUG/Projects/Pascal or http://www.jsoftware.com/jwiki/Essays/Pascal's_Triangle for more.

Re: Pascal's Triangle
 

That's because J is deliberately cryptic and hides all the hard work behind pre-written combinators and functions, in this case ! being "choose". Given the equivalent of these two defined in Haskell:

comb n k = foldl' (\x y -> ((n - x + 1) * y) `div` x) 1 [1..k]



table f xs ys = foldr ((:) . (`map` ys) . f) [] xs

Then producing Pascal's triangle is as simple as

join (table comb) [0..]

.

But overall a much simpler solution is

iterate (\xs -> zipWith (+) (0:xs) (xs++[0])) [1]

Re: Pascal's Triangle
 

I suppose anything is cryptic if you make no effort to understand it.

Even so, your statement is very odd. Apparently you are under the mis-apprehension that your failure to understand is the fault of a succinct, powerful notation and this means that the language was "deliberately" designed to be hard to understand.

Why would you even think something like this? I mean, why would someone deliberately make something difficult to understand? Most people I know strive for clarity. Do you think that "1+1" is cryptic because it uses a symbol "+" instead of the word "plus"? I suppose it is if you don't know math.

The other odd thing you seem to imply is that avoiding hard, unnecessary work is a bad thing. So, you're against the idea of carefully designed, well-tested, powerful, pre-written functions that you can combine in a clear, logically consistent manner?

This is all the more puzzling as your initial answer

join (table comb) [0..]

apparently very closely resembles the J solution of

 i. !/ i. 

because each "i." generates a vector of consecutive integers and "/" - similarly to (table comb) - combines these vectors in a table using "!". The "!" function, with a right and left arguments x and y returns the number of ways x things can be chosen from y possibilities - pretty much a direct statement of what you are doing when determining the coefficient of a polynomial expansion.

There are, of course other ways to do this more algorithmically and less mathematically as you appear to show in your final example. I say "appear" because that example is rather cryptic insofar as it uses the pre-written function "zipWith".

In J, I could write something I presume is similar to this:

   pascalNewRow=: 3 : '1,(2+/\y),1'  

based on the following rule for generating a new row of the triangle from the previous one: start and end with 1; add together adjacent pairs for the middle section. The above expression "2+/\" adds together each adjacent 2 numbers across the first dimension of an array; the "3 :" defines a function. Once you know that, the code is a straightforward expression of the rule.

Similarly to the final expression in Haskell, we iterate the function "pascalNewRow" by using an iterator expressed by some symbols, "^:" instead of the word "iterate". So, the expression

   pascalNewRow^:10 ] 1 

returns the first ten rows of Pascal's triangle (starting with a "seed" of 1).

I find this much less cryptic than the Haskell expression but, of course, I've made the effort to understand it.

In any case, this has wandered far off topic. Why don't I conclude by giving a solution to the original problem? Taking my J code as a template, start by defining a function to add adjacent pairs of a vector:

int *addAdjacentPairs( int vlen, int *vec)

{   int vc, win2[2];

    win2[0]=0;

    win2[1]=vec[0];

    for(vc=0;vc<=vlen;vc++)

    {   vec[vc]=win2[0]+win2[1];

        win2[0]=win2[1];

        win2[1]=vec[vc+1];

    }

// Danger: returned vector is 1 longer than input: allocate accordingly.

    return 0;

}

and simply use it in a loop:

// PascalTriangle.c: print some rows of Pascal's triangle.



#include<stdio.h>



int main(void)

{   int vec[21];

    int ii,vc,nn;

    printf("How many lines (up to 20) do you want? ");

    scanf("%d",&nn);

// Should verify result and handle errors but this is why I don't like these

// languages that don't handle arrays natively so I won't bother.

    printf("\n");



    for(ii=0;ii<nn;ii++)

    {   vec[ii]=1;

        for(vc=0;vc<=ii;vc++) printf("%i ",vec[vc]);

        printf("\n");

        addAdjacentPairs( ii, vec);

     }

    return 0;

}

This provides a nice example of how solving the problem with a high-level notation leads to a more elegant solution even in a lower-level language.