Provided by: libmath-planepath-perl_129-1_all 
NAME
Math::PlanePath::PowerArray -- array by powers
SYNOPSIS
use Math::PlanePath::PowerArray;
my $path = Math::PlanePath::PowerArray->new (radix => 2);
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is a split of N into an odd part and power of 2,
12 | 25 50 100 200 400 800 1600 3200 6400
11 | 23 46 92 184 368 736 1472 2944 5888
10 | 21 42 84 168 336 672 1344 2688 5376
9 | 19 38 76 152 304 608 1216 2432 4864
8 | 17 34 68 136 272 544 1088 2176 4352
7 | 15 30 60 120 240 480 960 1920 3840
6 | 13 26 52 104 208 416 832 1664 3328
5 | 11 22 44 88 176 352 704 1408 2816
4 | 9 18 36 72 144 288 576 1152 2304
3 | 7 14 28 56 112 224 448 896 1792
2 | 5 10 20 40 80 160 320 640 1280
1 | 3 6 12 24 48 96 192 384 768
Y=0 | 1 2 4 8 16 32 64 128 256
+------------------------------------------------------
X=0 1 2 3 4 5 6 7 8
For N=odd*2^k, the coordinates are X=k, Y=(odd-1)/2. The X coordinate is how many factors
of 2 can be divided out. The Y coordinate counts odd integers 1,3,5,7,etc as 0,1,2,3,etc.
This is clearer by writing N values in binary,
N values in binary
6 | 1101 11010 110100 1101000 11010000 110100000
5 | 1011 10110 101100 1011000 10110000 101100000
4 | 1001 10010 100100 1001000 10010000 100100000
3 | 111 1110 11100 111000 1110000 11100000
2 | 101 1010 10100 101000 1010000 10100000
1 | 11 110 1100 11000 110000 1100000
Y=0 | 1 10 100 1000 10000 100000
+---------------------------------------------------------
X=0 1 2 3 4 5
Column X=0 is all the odd numbers, column X=1 is exactly one low 0-bit, and so on.
Radix
The "radix" parameter can do the same dividing out in a higher base. For example radix 3
divides out factors of 3,
radix => 3
8 | 13 39 117 351 1053 3159 9477 28431
7 | 11 33 99 297 891 2673 8019 24057
6 | 10 30 90 270 810 2430 7290 21870
5 | 8 24 72 216 648 1944 5832 17496
4 | 7 21 63 189 567 1701 5103 15309
3 | 5 15 45 135 405 1215 3645 10935
2 | 4 12 36 108 324 972 2916 8748
1 | 2 6 18 54 162 486 1458 4374
Y=0 | 1 3 9 27 81 243 729 2187
+------------------------------------------------
X=0 1 2 3 4 5 6 7
N=1,3,9,27,etc on the X axis is the powers of 3.
N=1,2,4,5,7,etc on the Y axis is the integers N = 1or2 mod 3, ie. those not a multiple of
3. Notice if Y = 1or2 mod 4 then the N values in that row are all even, or if Y = 0or3
mod 4 then the N values are all odd.
radix => 3, N values in ternary
6 | 101 1010 10100 101000 1010000 10100000
5 | 22 220 2200 22000 220000 2200000
4 | 21 210 2100 21000 210000 2100000
3 | 12 120 1200 12000 120000 1200000
2 | 11 110 1100 11000 110000 1100000
1 | 2 20 200 2000 20000 200000
Y=0 | 1 10 100 1000 10000 100000
+----------------------------------------------------
X=0 1 2 3 4 5
Boundary Length
The points N=1 to N=2^k-1 inclusive have a boundary length
boundary = 2^k + 2k = 4,8,14,24,42,76,... (OEIS A100314)
For example N=1 to N=7 is
+---+
| 7 |
+ +
| 5 |
+ +---+
| 3 6 |
+ +---+
| 1 2 4 |
+---+---+---+
The height is the odd numbers, so 2^(k-1). The width is the power k. So total boundary
2*height+2*width = 2^k + 2k.
If N=2^k is included then it's on the X axis and so add 2 for boundary = 2^k + 2k + 2
(OEIS 2*A052968).
For another radix the calculation is similar
boundary = 2 * (radix-1) * radix^(k-1) + 2*k
For example radix=3, N=1 to N=8 is
8
7
5
4
2 6
1 3
The height is the non-multiples of the radix, so (radix-1)/radix * radix^k. The width is
the power k. Total boundary = 2*height+2*width.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
"$path = Math::PlanePath::PowerArray->new ()"
Create and return a new path object.
"($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number $n on the path. Points begin at 1 and if
"$n < 0" then the return is an empty list.
"$n = $path->xy_to_n ($x,$y)"
Return the N point number at coordinates "$x,$y". If "$x<0" or "$y<0" then there's no
N and the return is "undef".
N values grow rapidly with $x. Pass in a number type such as "Math::BigInt" to
preserve precision.
"($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.
FORMULAS
Rectangle to N Range
Within each row, increasing X is increasing N. Within each column, increasing Y is
increasing N. So in a rectangle the lower left corner is the minimum N and the upper
right is the maximum N.
| N max
| ----------+
| | ^ |
| | | |
| | ----> |
| +----------
| N min
+-------------------
N to Turn Left or Right
The turn left or right is given by
radix = 2 left at N==0 mod radix and N==1mod4, right otherwise
radix >= 3 left at N==0 mod radix
right at N=1 or radix-1 mod radix
straight otherwise
The points N!=0 mod radix are on the Y axis and those N==0 mod radix are off the axis.
For that reason the turn at N==0 mod radix is to the left,
|
C--
---
A--__ -- point B is N=0 mod radix,
| --- B turn left A-B-C is left
For radix>=3 the turns at A and C are to the right, since the point before A and after C
is also on the Y axis. For radix>=4 there's of run of points on the Y axis which are
straight.
For radix=2 the "B" case N=0 mod 2 applies, but for the A,C points in between the turn
alternates left or right.
1-- N=1 mod 4 3-- N=3 mod 4
\ -- turn left \ -- turn right
\ -- \ --
2 -- 2 --
-- --
-- --
0 4
Points N=2 mod 4 are X=1 and Y=N/2 whereas N=0 mod 4 has 2 or more trailing 0 bits so X>1
and Y<N/2.
N mod 4 turn
------- ------
0 left for radix=2
1 left
2 left
3 right
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A007814> (etc)
radix=2
A007814 X coordinate, count low 0-bits of N
A006519 2^X
A025480 Y coordinate of N-1, ie. seq starts from N=0
A003602 Y+1, being k for which N=(2k-1)*2^m
A153733 2*Y of N-1, strip low 1 bits
A000265 2*Y+1, strip low 0 bits
A094267 dX, change count low 0-bits
A050603 abs(dX)
A108715 dY, change in Y coordinate
A000079 N on X axis, powers 2^X
A057716 N not on X axis, the non-powers-of-2
A005408 N on Y axis (X=0), the odd numbers
A003159 N in X=even columns, even trailing 0 bits
A036554 N in X=odd columns
A014480 N on X=Y diagonal, (2n+1)*2^n
A118417 N on X=Y+1 diagonal, (2n-1)*2^n
(just below X=Y diagonal)
A054582 permutation N by diagonals, upwards
A209268 inverse
A135764 permutation N by diagonals, downwards
A249725 inverse
A075300 permutation N-1 by diagonals, upwards
A117303 permutation N at transpose X,Y
A100314 boundary length for N=1 to N=2^k-1 inclusive
being 2^k+2k
A131831 same, after initial 1
A052968 half boundary length N=1 to N=2^k inclusive
being 2^(k-1)+k+1
radix=3
A007949 X coordinate, power-of-3 dividing N
A000244 N on X axis, powers 3^X
A001651 N on Y axis (X=0), not divisible by 3
A016051 N on Y column X=1
A051063 N on Y column X=1
A007417 N in X=even columns, even trailing 0 digits
A145204 N in X=odd columns (extra initial 0)
A141396 permutation, N by diagonals down from Y axis
A191449 permutation, N by diagonals up from X axis
A135765 odd N by diagonals, deletes the Y=1,2mod4 rows
A000975 Y at N=2^k, being binary "10101..101"
radix=4
A000302 N on X axis, powers 4^X
radix=5
A112765 X coordinate, power-of-5 dividing N
A000351 N on X axis, powers 5^X
radix=6
A122841 X coordinate, power-of-6 dividing N
radix=10
A011557 N on X axis, powers 10^X
A067251 N on Y axis, not a multiple of 10
A151754 Y coordinate of N=2^k, being floor(2^k*9/10)
SEE ALSO
Math::PlanePath, Math::PlanePath::WythoffArray, Math::PlanePath::ZOrderCurve
David M. Bradley "Counting Ordered Pairs", Mathematics Magazine, volume 83, number 4,
October 2010, page 302, DOI 10.4169/002557010X528032.
<http://www.math.umaine.edu/~bradley/papers/JStor002557010X528032.pdf>
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it under the terms
of the GNU General Public License as published by the Free Software Foundation; either
version 3, or (at your option) any later version.
Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with Math-
PlanePath. If not, see <http://www.gnu.org/licenses/>.