Math::PlanePath::PyramidRows(3pm)

Step Parameter

A "step" parameter controls how much wider each row is than the preceding, to make wider pyramids. For example step 4

    my $path = Math::PlanePath::PyramidRows->new (step => 4);

makes each row 2 wider on each side successively

   29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45      4
         16 17 18 19 20 21 22 23 24 25 26 27 28            3
                7  8  9 10 11 12 13 14 15                  2
                      2  3  4  5  6                        1
                            1                        <-  Y=0
         -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6 ...

If the step is an odd number then the extra is at the right, so step 3 gives

    13  14  15  16  17  18  19  20  21  22       3
         6   7   8   9  10  11  12               2
             2   3   4   5                       1
                 1                         <-  Y=0
    -3  -2  -1  X=0  1   2   3   4 ...

Or step 1 goes solely to the right. This is equivalent to the Diagonals path, but columns shifted up to make horizontal rows.

    11  12  13  14  15         4
     7   8   9  10             3       step => 1
     4   5   6                 2
     2   3                     1
     1                   <-  Y=0
    X=0  1   2   3   4 ...

Step 0 means simply a vertical, each row 1 wide and not increasing. This is consistent but unlikely to be much use. The Rows path with "width" 1 does this too.

     5         4
     4         3     step => 0
     3         2
     2         1
     1    <- Y=0
    X=0

Various number sequences fall in regular patterns positions depending on the step. Large steps are not particularly interesting and quickly become very wide. A limit might be desirable in a user interface, but there's no limit in the code as such.

Align Parameter

An optional "align" parameter controls how the points are arranged relative to the Y axis. The default shown above is "centre".

"right" means points to the right of the axis,

    align=>"right"
    26  27  28  29  30  31  32  33  34  35  36        5
    17  18  19  20  21  22  23  24  25                4
    10  11  12  13  14  15  16                        3
     5   6   7   8   9                                2
     2   3   4                                        1
     1                                            <- Y=0
    X=0  1   2   3   4   5   6   7   8   9  10

"left" is similar but to the left of the Y axis, ie. into negative X.

    align=>"left"
    26  27  28  29  30  31  32  33  34  35  36        5
            17  18  19  20  21  22  23  24  25        4
                    10  11  12  13  14  15  16        3
                             5   6   7   8   9        2
                                     2   3   4        1
                                             1    <- Y=0
    -10 -9  -8  -7  -6  -5  -4  -3  -2  -1  X=0

The step parameter still controls how much longer each row is than its predecessor.

N Start

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

    n_start => 0
    16 17 18 19 20 21 22 23 24        4
        9 10 11 12 13 14 15           3
           4  5  6  7  8              2
              1  2  3                 1
                 0                <- Y=0
    --------------------------
    -4 -3 -2 -1 X=0 1  2  3  4

Step 3 Pentagonals

For step=3 the pentagonal numbers 1,5,12,22,etc, P(k) = (3k-1)*k/2, are at the rightmost end of each row. The second pentagonal numbers 2,7,15,26, S(k) = (3k+1)*k/2 are the vertical at x=-1. Those second numbers are obtained by P(-k), and the two together are the "generalized pentagonal numbers".

Both these sequences are composites from 12 and 15 onwards, respectively, and the immediately preceding P(k)-1, P(k)-2, and S(k)-1, S(k)-2 are too. They factorize simply as

    P(k)   = (3*k-1)*k/2
    P(k)-1 = (3*k+2)*(k-1)/2
    P(k)-2 = (3*k-4)*(k-1)/2
    S(k)   = (3*k+1)*k/2
    S(k)-1 = (3*k-2)*(k+1)/2
    S(k)-2 = (3*k+4)*(k-1)/2

Plotting the primes on a step=3 "PyramidRows" has the second pentagonal S(k),S(k)-1,S(k)-2 as a 3-wide vertical gap of no primes at X=-1,-2,-3. The the plain pentagonal P(k),P(k-1),P(k)-2 are the endmost three N of each row non-prime. The vertical is much more noticeable in a plot.

       no primes these three columns         no primes these end three
         except the low 2,7,13                     except low 3,5,11
               |  |  |                                /  /  /
     52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
        36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
           23 24 25 26 27 28 29 30 31 32 33 34 35
              13 14 15 16 17 18 19 20 21 22
                  6  7  8  9 10 11 12
                     2  3  4  5
                        1
     -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9 10 11 ...

With align="left" the end values can be put into columns,

                                no primes these end three
    align => "left"                  except low 3,5,11
                                            |  |  |
    36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51        5
             23 24 25 26 27 28 29 30 31 32 33 34 35        4
                      13 14 15 16 17 18 19 20 21 22        3
                                6  7  8  9 10 11 12        2
                                         2  3  4  5        1
                                                  1    <- Y=0
              ... -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0

In general a constant offset S(k)-c is a column and from P(k)-c is a diagonal sloping up dX=2,dY=1 right. The simple factorizations above using the roots of the quadratic P(k)-c or S(k)-c is possible whenever 24*c+1 is a perfect square. This means the further columns S(k)-5, S(k)-7, S(k)-12, etc also have no primes.

The columns S(k), S(k)-1, S(k)-2 are prominent because they're adjacent. There's no other adjacent columns of this type because the squares after 49 are too far apart for 24*c+1 to be a square for successive c. Of course there could be other reasons for other columns or diagonals to have few or many primes.

Descriptive Methods

"$x = $path->sumxy_minimum()"

"$x = $path->sumxy_maximum()"

Return the minimum or maximum values taken by coordinate sum X+Y reached by integer N values in the path. If there's no minimum or maximum then return "undef".

The path is right and above the X=-Y diagonal, thus giving a minimum sum, in the following cases.

    align      condition for sumxy_minimum=0
    ------     -----------------------------
    centre              step <= 3
    right               always
    left                step <= 1

"$x = $path->diffxy_minimum()"

"$x = $path->diffxy_maximum()"

Return the minimum or maximum values taken by coordinate difference X-Y reached by integer N values in the path. If there's no minimum or maximum then return "undef".

The path is left and above the X=Y leading diagonal, thus giving a minimum X-Y difference, in the following cases.

    align      condition for diffxy_minimum=0
    ------     -----------------------------
    centre              step <= 2
    right               step <= 1
    left                always