`pascal.raku`

Pascal's triangle & what it encodes.

Builds 12 rows of Pascal's triangle — each row grows from the one above by adding neighbouring entries with the zip operator Z+ — and prints it centred. It then checks row 11 against the binomial coefficients C(11, k) computed directly, and redraws the triangle keeping only the odd entries, which trace out the Sierpinski fractal.

The program

Edit it and press Run — it executes in your browser.

#!/usr/bin/env raku
# Pascal's triangle, and two things it secretly encodes.
#
# Shows off list building (each row grows from the one above by adding
# neighbours with the zip operator `Z+`), combinatorics, and a little
# ASCII formatting. Every entry is an exact arbitrary-precision `Int`.

my $rows = 12;

# Build the triangle. The next row is 1, the pairwise sums of the current
# row's neighbours, then 1 — which is exactly what `Z+` (zip-with-plus)
# gives when a row is added to itself shifted by one.
my @triangle;
my @row = 1;
for ^$rows {
    @triangle.push: @row.clone;   # snapshot — `@row = …` below refills the same container
    @row = 1, |(@row Z+ @row[1 .. *]), 1;
}

# Print it centred. Each entry sits in a fixed-width cell, and every row
# is indented by half a cell so the triangle stays symmetric.
my $cell = @triangle[*-1].map(*.chars).max + 1;
say "Pascal's triangle:";
for @triangle -> @r {
    my $indent = ' ' x ($cell * ($rows - @r.elems) div 2);
    say $indent ~ @r.map({ sprintf('%*d', $cell, $_) }).join;
}

# The entry in row n, position k is the binomial coefficient "n choose k".
# Compute one directly and check it against the triangle we built.
sub choose($n, $k) { ([*] $n - $k + 1 .. $n) div [*] 1 .. $k }

say '';
say 'Row 11 is the binomial coefficients C(11, k):';
say '  from triangle: ', @triangle[11];
say '  from C(11, k): ', (0 .. 11).map({ choose(11, $_) });
say '  match: ', @triangle[11].List eqv (0 .. 11).map({ choose(11, $_) }).List;

# Colour each entry by parity: the odd numbers alone trace out the
# Sierpinski triangle, a fractal hiding inside the arithmetic.
say '';
say 'Odd entries only (the Sierpinski triangle):';
for @triangle -> @r {
    my $indent = ' ' x ($rows - @r.elems);
    say $indent ~ @r.map({ $_ % 2 ?? '#' !! ' ' }).join(' ');
}

Open in the playground ↗ Source on GitHub ↗

Output

Pascal's triangle:
                         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
         1   8  28  56  70  56  28   8   1
       1   9  36  84 126 126  84  36   9   1
     1  10  45 120 210 252 210 120  45  10   1
   1  11  55 165 330 462 462 330 165  55  11   1

Row 11 is the binomial coefficients C(11, k):
  from triangle: [1 11 55 165 330 462 462 330 165 55 11 1]
  from C(11, k): (1 11 55 165 330 462 462 330 165 55 11 1)
  match: True

Odd entries only (the Sierpinski triangle):
           #
          # #
         #   #
        # # # #
       #       #
      # #     # #
     #   #   #   #
    # # # # # # # #
   #               #
  # #             # #
 #   #           #   #
# # # #         # # # #

Feature focus: combinatorics, Z+, formatting.