# Pascal’s Triangle

One of the exercises in Structure and Implementation of Computer Programs deals with generating elements of the Pascal’s Triangle.

The following pattern of numbers is called Pascal’s Triangle

``````    1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
...``````

The numbers at the edge of the triangle are 1, and each number inside is the sum of two numbers above it. The exercise asks us to find the elements of Pascal’s triangle by means of a recursive process.

I find it helpful to de-skew the triangle so that the rows and columns line up visually. Here’s the modified triangle:

``````1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
...``````

I decided to write the solution in Haskell, and here it is:

``````-- A: Function that finds the element of Pascal's Triangle for a given row & column
pascal :: Int -> Int -> Int
pascal 0 _ = 1
pascal _ 0 = 1
pascal row col
| row == col = 1
| col > row  = error "Column can't be greater than row"
| otherwise  = (pascal (row-1) (col-1)) + (pascal (row-1) col)``````

This code is great for generating the element at a given row and column (e.g. `pascal 3 2` produces `3`), but not so great if we wanted to generate the entire triangle up to `n` rows, as recursion would cause the code to generate the elements at row `n-1` multiple times.

In order to generate the entire triangle, it is better to do it via an iterative process as follows: generate the first row, and from that generate the second row, and so on. I other words, we are generating the next row of the triangle, by adding each pair in the current row. My solution is as follows:

``````-- Takes an array of integers, and adds pairs of numbers to create a new
-- array. For example, [1,2,3,4] changes to [(1+2), (2+3), (3+4)], resulting
-- in [3,5,7]
sumOfPairs :: [Integer] -> [Integer]
sumOfPairs [] = []
sumOfPairs (x:[]) = []
sumOfPairs (x:y:ys) = (x+y) : sumOfPairs (y:ys)

-- Given a row of Pascal's triangle, returns the next row
next :: [Integer] -> [Integer]
next xs = 1 :  (sumOfPairs xs) ++ [1]

-- Generates the Pascal's triangle. This function generates an
-- infinite triangle, so its usage needs to be coupled with the
-- take command. For example: take 5 pascalTriangle
pascalTriangle :: [[Integer]]
pascalTriangle = pascalHelper [1]
where pascalHelper xs = xs : pascalHelper (next xs)``````

Here are a couple of examples of generating the triangle:

``````*Main> take 4 pascalTriangle
[[1],[1,1],[1,2,1],[1,3,3,1]]

-- Using mapM_ with print to 'pretty-print' the triangle
*Main> mapM_ print \$ take 8 pascalTriangle
[1]
[1,1]
[1,2,1]
[1,3,3,1]
[1,4,6,4,1]
[1,5,10,10,5,1]
[1,6,15,20,15,6,1]
[1,7,21,35,35,21,7,1]``````

The function `pascalTriangle` can be made more concise by using the built-in `iterate` instead of my `pascalHelper`:

``````pascalTriangle :: [[Integer]]
pascalTriangle = iterate next [1]``````

By this point, I was reasonably satisfied with the code. However, being relatively new to Haskell, I decided to explore and see if there were other (and possibly better and more idiomatic) ways to solve this problem. My favorite was by Neil Mitchell, who solved the problem in two lines! I encourage you to go read Neil’s entire post. Here’s the two-line version of Haskell’s triangle:

``````-- By: Neil Mitchell

next xs = zipWith (+) ([0] ++ xs) (xs ++ [0])
pascal = iterate next [1]``````

In my view, this code is very elegant and beautiful and does a great job of highlighting the power of Haskell.