 Math::PlanePath::PyramidRows(3) points stacked up in a pyramid

## SYNOPSIS

use Math::PlanePath::PyramidRows;
my \$path = Math::PlanePath::PyramidRows->new;
my (\$x, \$y) = \$path->n_to_xy (123);

## DESCRIPTION

This path arranges points in successively wider rows going upwards so as to form an upside-down pyramid. The default step is 2, ie. each row 2 wider than the preceding, an extra point at the left and the right,

```    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
-4  -3  -2  -1  X=0  1   2   3   4 ...
```

The right end N=1,4,9,16,etc is the perfect squares. The vertical 2,6,12,20,etc at x=-1 is the pronic numbers s*(s+1), half way between those successive squares.

The step 2 is the same as the "PyramidSides", "Corner" and "SacksSpiral" paths. For the "SacksSpiral", spiral arms going to the right correspond to diagonals in the pyramid, and arms to the left correspond to verticals.

## 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.

```    step => 1
11  12  13  14  15                4
7   8   9  10                    3
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 unlikely to be much use. The Rows path with "width" 1 does this too.

```    step => 0
5        4
4        3
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.

## FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"\$path = Math::PlanePath::PyramidRows->new ()"
"\$path = Math::PlanePath::PyramidRows->new (step => \$integer, align => \$str, n_start => \$n)"
Create and return a new path object. The default "step" is 2. "align" is a string, one of

```    "centre"    the default
"right"     points aligned right of the Y axis
"left"      points aligned left of the Y axis
```

Points are always numbered from left to right in the rows, the alignment changes where each row begins (or ends).

"(\$x,\$y) = \$path->n_to_xy (\$n)"
Return the X,Y coordinates of point number \$n on the path.

For "\$n <= 0" the return is an empty list since the path starts at N=1.

"\$n = \$path->xy_to_n (\$x,\$y)"
Return the point number for coordinates "\$x,\$y". \$x and \$y are each rounded to the nearest integer, which has the effect of treating each point in the pyramid as a square of side 1. If "\$x,\$y" is outside the pyramid the return is "undef".
"(\$n_lo, \$n_hi) = \$path->rect_to_n_range (\$x1,\$y1, \$x2,\$y2)"
The returned range is exact, meaning \$n_lo and \$n_hi are the smallest and biggest in the rectangle.

## 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
```

## OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A023531> (etc)

```    step=1
A002262    X coordinate, runs 0 to k
A003056    Y coordinate, k repeated k+1 times
A051162    X+Y sum
A025581    Y-X diff, runs k to 0
A079904    X*Y product
A069011    X^2+Y^2, n_to_rsquared()
A080099    X bitwise-AND Y
A080098    X bitwise-OR  Y
A051933    X bitwise-XOR Y
A050873    GCD(X+1,Y+1) greatest common divisor by rows
A051173    LCM(X+1,Y+1) least common multiple by rows
A023531    dY, being 1 at triangular numbers (but starting n=0)
A167407    dX-dY, change in X-Y (extra initial 0)
A129184    turn 1=left, 0=right or straight
A079824    N total along each opposite diagonal
A000124    N on Y axis (triangular+1)
A000217    N on X=Y diagonal, extra initial 0
step=1, n_start=0
A109004    GCD(X,Y) greatest common divisor starting (0,0)
A103451    turn 1=left or right,0=straight, but extra initial 1
A103452    turn 1=left,0=straight,-1=right, but extra initial 1
step=2
A196199    X coordinate, runs -n to +n
A000196    Y coordinate, n appears 2n+1 times
A053186    X+Y, being distance to next higher square
A010052    dY,  being 1 at perfect square row end
A000290    N on X=Y diagonal, extra initial 0
A002522    N on X=-Y North-West diagonal (start row), Y^2+1
A004201    N for which X>=0, ie. right hand half
A020703    permutation N at -X,Y
step=2, n_start=0
A005563    N on X=Y diagonal, Y*(Y+2)
A000290    N on X=-Y North-West diagonal (start row), Y^2
step=2, n_start=2
A059100    N on north-west diagonal (start each row), Y^2+2
A053615    abs(X), runs k..0..k
step=2, align=right, n_start=0
A196199    X-Y, runs -k to +k
A053615    abs(X-Y), runs k..0..k
step=2, align=left, n_start=0
A005563    N on Y axis, Y*(Y+2)

step=3
A180447    Y coordinate, n appears 3n+1 times
A104249    N on Y axis, Y*(3Y+1)/2+1
A143689    N on X=-Y North-West diagonal
step=3, n_start=0
A005449    N on Y axis, second pentagonals Y*(3Y+1)/2
A000326    N on diagonal north-west, pentagonals Y*(3Y-1)/2
step=4
A084849    N on Y axis
A001844    N on X=Y diagonal (North-East)
A058331    N on X=-Y North-West diagonal
A221217    permutation N at -X,Y
step=4, n_start=0
A014105    N on Y axis, the second hexagonal numbers
A046092    N on X=Y diagonal, 4*triangular numbers
step=4, align=right, n_start=0
A060511    X coordinate, amount n exceeds hexagonal number
A000384    N on Y axis, the hexagonal numbers
A001105    N on X=Y diagonal, 2*squares
step=5
A116668    N on Y axis
step=6
A056108    N on Y axis
A056109    N on X=Y diagonal (North-East)
A056107    N on X=-Y North-West diagonal
step=8
A053755    N on X=-Y North-West diagonal
step=9
A006137    N on Y axis
A038764    N on X=Y diagonal (North-East)
```

<http://user42.tuxfamily.org/math-planepath/index.html>