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    Nright = ((3Y+8)*Y + 4)/4    if Y even
             ((3Y+8)*Y + 5)/4    if Y odd

           = Nleft(Y+1) - 1   ie. 1 before next Nleft

The row width Xmax-Xmin = 2*Y but with the gaps the number of visited points
in a row is less than that,

    rowpoints = 3*Y/2 + 1        if Y even
                3*(Y+1)/2        if Y odd

For any Y of course the Nleft to Nright difference is the number of points
in the row too

    rowpoints = Nright - Nleft + 1

=cut

# even Nright - Nleft + 1
#      = ((3Y+8)Y + 4)/4 - ((3Y+2)*Y + 4)/4 + 1
#      = [ (3Y+8)Y + 4 - (3Y+2)*Y - 4 ]/4 + 1
#      = [ (3Y+8)Y - (3Y+2)*Y ] / 4 + 1
#      = (3Y+8-3Y-2)Y/4 + 1
#      = 6Y/4 + 1
#      = 3Y/2 + 1
# odd Nright - Nleft + 1
#     = ((3Y+8)Y + 5)/4 - ((3Y+2)*Y + 3)/4 + 1
#     = [ (3Y+8)Y + 5 - (3Y+2)*Y - 3 ]/4 + 1
#     = [ (3Y+8)Y - (3Y+2)*Y + 2 ]/4 + 1
#     = [ 6Y + 2 ]/4 + 1
#     = [ 6Y + 2 + 4]/4
#     = [ 6Y + 6]/4
#     = 3(Y+1)/2

=head2 N Start

The default is to number points starting N=1 as shown above.  An optional
C<n_start> can give a different start, in the same pattern.  For example to
start at 0,

=cut

# math-image --path=CellularRule190,n_start=0 --all --output=numbers --size=75x6

=pod

    n_start => 0

    21 22 23    24 25 26    27 28 29          5 
       14 15 16    17 18 19    20             4 
           8  9 10    11 12 13                3 
              4  5  6     7                   2 
                 1  2  3                      1 
                    0                     <- Y=0

    -5 -4 -3 -2 -1 X=0 1  2  3  4  5

The effect is to push each N rightwards by 1, and wrapping around.  So the
N=0,1,4,8,14,etc on the left were on the right of the default n_start=1.
This also has the effect of removing the +1 in the Nright formula given
above, so

    Nleft = triangular(Y) + quartersquare(Y)

=head1 FUNCTIONS

See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.

=over 4

=item C<$path = Math::PlanePath::CellularRule190-E<gt>new ()>

=item C<$path = Math::PlanePath::CellularRule190-E<gt>new (mirror =E<gt> 1, n_start =E<gt> $n)>

Create and return a new path object.

=item C<($x,$y) = $path-E<gt>n_to_xy ($n)>

Return the X,Y coordinates of point number C<$n> on the path.

=item C<$n = $path-E<gt>xy_to_n ($x,$y)>

Return the point number for coordinates C<$x,$y>.  C<$x> and C<$y> are each
rounded to the nearest integer, which has the effect of treating each cell
as a square of side 1.  If C<$x,$y> is outside the pyramid or on a skipped
cell the return is C<undef>.

=item C<($n_lo, $n_hi) = $path-E<gt>rect_to_n_range ($x1,$y1, $x2,$y2)>

The returned range is exact, meaning C<$n_lo> and C<$n_hi> are the smallest
and biggest in the rectangle.

=back

=head1 OEIS

This pattern is in Sloane's Online Encyclopedia of Integer Sequences in a
couple of forms,

=over

L<http://oeis.org/A037576> (etc)

=back

    A037576     whole-row used cells as bits of a bignum
    A265688       and in binary
    A071039     1/0 used and unused cells across rows
    A118111       same
    A071041     1/0 used and unused of mirrored rule 246 

    n_start=0
      A006578   N at left of each row (X=-Y),
                  and at right of each row when mirrored,
                  being triangular+quartersquare

=head1 SEE ALSO

L<Math::PlanePath>,
L<Math::PlanePath::CellularRule>,