# Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
# This file is part of MathPlanePath.
#
# MathPlanePath 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.
#
# MathPlanePath 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 MathPlanePath. If not, see .
# http://d4maths.lowtech.org/mirage/ulam.htm
# http://d4maths.lowtech.org/mirage/img/ulam.gif
# sample gif of primes made by APL or something
#
# http://www.sciencenews.org/view/generic/id/2696/title/Prime_Spirals
# Ulam's spiral of primes
#
# http://yoyo.cc.monash.edu.au/%7Ebunyip/primes/primeSpiral.htm
# http://yoyo.cc.monash.edu.au/%7Ebunyip/primes/triangleUlam.htm
# Pulchritudinous Primes of Ulam spiral.
# http://mathworld.wolfram.com/PrimeSpiral.html
#
# Mark C. ChuCarroll "The Surprises Never Eend: The Ulam Spiral of Primes"
# http://scienceblogs.com/goodmath/2010/06/the_surprises_never_eend_the_u.php
#
# http://yoyo.cc.monash.edu.au/%7Ebunyip/primes/index.html
# including image highlighting the lines
# S. M. Ellerstein, The square spiral, J. Recreational
# Mathematics 29 (#3, 1998) 188; 30 (#4, 19992000), 246250.
#
# Stein, M. and Ulam, S. M. "An Observation on the
# Distribution of Primes." Amer. Math. Monthly 74, 4344,
# 1967.
#
# Stein, M. L.; Ulam, S. M.; and Wells, M. B. "A Visual
# Display of Some Properties of the Distribution of Primes."
# Amer. Math. Monthly 71, 516520, 1964.
# cf sides alternately prime and fibonacci
# A160790 corner N
# A160791 side lengths, alternately integer and triangular adding that integer
# A160792 corner N
# A160793 side lengths, alternately integer and sum primes
# A160794 corner N
# A160795 side lengths, alternately primes and fibonaccis
package Math::PlanePath::SquareSpiral;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 116;
use Math::PlanePath;
use Math::PlanePath::Base::NSEW;
@ISA = ('Math::PlanePath::Base::NSEW',
'Math::PlanePath');
use Math::PlanePath::Base::Generic
'round_nearest';
# uncomment this to run the ### lines
#use Smart::Comments '###';
# Note: this shared by other paths
use constant parameter_info_array =>
[
{ name => 'wider',
display => 'Wider',
type => 'integer',
minimum => 0,
default => 0,
width => 3,
description => 'Wider path.',
},
Math::PlanePath::Base::Generic::parameter_info_nstart1(),
];
use constant xy_is_visited => 1;
# 2w+4  2w+3  w+2
#  
# 2w+5 0 w+1
# 
# 2w+6 
# ^
# X=0
#
sub x_negative_at_n {
my ($self) = @_;
return $self>n_start + ($self>{'wider'} ? 0 : 4);
}
sub y_negative_at_n {
my ($self) = @_;
return $self>n_start + 2*$self>{'wider'} + 6;
}
sub _UNDOCUMENTED__dxdy_list_at_n {
my ($self) = @_;
return $self>n_start + 2*$self>{'wider'} + 4;
}
#
sub new {
my $self = shift>SUPER::new (@_);
# parameters
$self>{'wider'} = 0; # default
if (! defined $self>{'n_start'}) {
$self>{'n_start'} = $self>default_n_start;
}
return $self;
}
# wider==0
# base from bottomright corner
# d = [ 1, 2, 3, 4 ]
# N = [ 2, 10, 26, 50 ]
# N = (4 d^2  4 d + 2)
# d = 1/2 + sqrt(1/4 * $n + 4/16)
#
# wider==1
# base from bottomright corner
# d = [ 1, 2, 3, 4 ]
# N = [ 3, 13, 31, 57 ]
# N = (4 d^2  2 d + 1)
# d = 1/4 + sqrt(1/4 * $n + 3/16)
#
# wider==2
# base from bottomright corner
# d = [ 1, 2, 3, 4 ]
# N = [ 4, 16, 36, 64 ]
# N = (4 d^2)
# d = 0 + sqrt(1/4 * $n + 0)
#
# wider==3
# base from bottomright corner
# d = [ 1, 2, 3 ]
# N = [ 5, 19, 41 ]
# N = (4 d^2 + 2 d  1)
# d = 1/4 + sqrt(1/4 * $n + 5/16)
#
# N = 4*d^2 + (4+2*w)*d + (2w)
# = 4*$d*$d + (4+2*$w)*$d + (2$w)
# d = 1/2w/4 + sqrt(1/4*$n + b^24ac)
# (b^24ac)/(2a)^2 = [ (2w4)^2  4*4*(2w) ] / 64
# = [ 4w^2  16w + 16  32 + 16w ] / 64
# = [ 4w^2  16 ] / 64
# = [ w^2  4 ] / 16
# d = 1/2w/4 + sqrt(1/4*$n + (w^2  4) / 16)
# = 1/4 * (2w + sqrt(4*$n + w^2  4))
# = 0.25 * (2$w + sqrt(4*$n + $w*$w  4))
#
# then offset the base by +4*$d+$w1 for top left corner for +/ remainder
# rem = $n  (4*$d*$d + (4+2*$w)*$d + (2$w) + 4*$d + $w  1)
# = $n  (4*$d*$d + (4+2*$w)*$d + 2  $w + 4*$d + $w  1)
# = $n  (4*$d*$d + (4+2*$w)*$d + 1  $w + 4*$d + $w)
# = $n  (4*$d*$d + (4+2*$w)*$d + 1 + 4*$d)
# = $n  (4*$d*$d + (2*$w)*$d + 1)
# = $n  ((4*$d + 2*$w)*$d + 1)
#
sub n_to_xy {
my ($self, $n) = @_;
#### SquareSpiral n_to_xy: $n
$n = $n  $self>{'n_start'}; # starting $n==0, warn if $n==undef
if ($n < 0) {
#### before n_start ...
return;
}
my $w = $self>{'wider'};
my $w_right = int($w/2);
my $w_left = $w  $w_right;
if ($n <= $w+1) {
#### centre horizontal
# n=0 at w_left
# x = $n  int(($w+1)/2)
# = $n  int(($w+1)/2)
return ($n  $w_left, # n=0 at w_left
0);
}
my $d = int ((2$w + sqrt(int(4*$n) + $w*$w)) / 4);
#### d frac: ((2$w + sqrt(int(4*$n) + $w*$w)) / 4)
#### $d
#### base: 4*$d*$d + (4+2*$w)*$d + (2$w)
$n = ((4*$d + 2*$w)*$d);
#### remainder: $n
if ($n >= 0) {
if ($n <= 2*$d) {
### left vertical
return ($d  $w_left,
$n + $d);
} else {
### bottom horizontal
return ($n  $w_left  3*$d,
$d);
}
} else {
if ($n >= 2*$d$w) {
### top horizontal
return ($n  $d  $w_left,
$d);
} else {
### right vertical
return ($d + $w_right,
$n + 3*$d + $w);
}
}
}
sub xy_to_n {
my ($self, $x, $y) = @_;
my $w = $self>{'wider'};
my $w_right = int($w/2);
my $w_left = $w  $w_right;
$x = round_nearest ($x);
$y = round_nearest ($y);
### xy_to_n: "x=$x, y=$y"
### $w_left
### $w_right
my $d;
if (($d = $x  $w_right) > abs($y)) {
### right vertical
### $d
#
# base bottom right per above
### BR: 4*$d*$d + (4+2*$w)*$d + (2$w)
# then +$d1 for the y=0 point
# N_Y0 = 4*$d*$d + (4+2*$w)*$d + (2$w) + $d1
# = 4*$d*$d + (3+2*$w)*$d + (2$w) + 1
# = 4*$d*$d + (3+2*$w)*$d + 1$w
### N_Y0: (4*$d + 3 + 2*$w)*$d + 1$w
#
return (4*$d + 3 + 2*$w)*$d  $w + $y + $self>{'n_start'};
}
if (($d = $x  $w_left) > abs($y)) {
### left vertical
### $d
#
# top left per above
### TL: 4*$d*$d + (2*$w)*$d + 1
# then +$d for the y=0 point
# N_Y0 = 4*$d*$d + (2*$w)*$d + 1 + $d
# = 4*$d*$d + (1 + 2*$w)*$d + 1
### N_Y0: (4*$d + 1 + 2*$w)*$d + 1
#
return (4*$d + 1 + 2*$w)*$d  $y + $self>{'n_start'};
}
$d = abs($y);
if ($y > 0) {
### top horizontal
### $d
#
# top left per above
### TL: 4*$d*$d + (2*$w)*$d + 1
# then ($d+$w_left) for the x=0 point
# N_X0 = 4*$d*$d + (2*$w)*$d + 1 + ($d+$w_left)
# = 4*$d*$d + (1 + 2*$w)*$d + 1  $w_left
### N_Y0: (4*$d  1 + 2*$w)*$d + 1  $w_left
#
return (4*$d  1 + 2*$w)*$d  $w_left  $x + $self>{'n_start'};
}
### bottom horizontal, and centre y=0
### $d
#
# top left per above
### TL: 4*$d*$d + (2*$w)*$d + 1
# then +2*$d to bottom left, +$d+$w_left for the x=0 point
# N_X0 = 4*$d*$d + (2*$w)*$d + 1 + 2*$d + $d+$w_left)
# = 4*$d*$d + (3 + 2*$w)*$d + 1 + $w_left
### N_Y0: (4*$d + 3 + 2*$w)*$d + 1 + $w_left
#
return (4*$d + 3 + 2*$w)*$d + $w_left + $x + $self>{'n_start'};
}
# hi is exact but lo is not
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
$x1 = round_nearest ($x1);
$y1 = round_nearest ($y1);
$x2 = round_nearest ($x2);
$y2 = round_nearest ($y2);
# ENHANCEME: find actual minimum if rect doesn't cover 0,0
return ($self>{'n_start'},
max ($self>xy_to_n($x1,$y1),
$self>xy_to_n($x2,$y1),
$self>xy_to_n($x1,$y2),
$self>xy_to_n($x2,$y2)));
# my $w = $self>{'wider'};
# my $w_right = int($w/2);
# my $w_left = $w  $w_right;
#
# my $d = 1 + max (abs($y1),
# abs($y2),
# $x1  $w_right, $x1  $w_left,
# $x2  $w_right, $x2  $w_left,
# 1);
# ### $d
# ### is: $d*$d
#
# # ENHANCEME: find actual minimum if rect doesn't cover 0,0
# return (1,
# (4*$d  4 + 2*$w)*$d + 2); # bottomright
}
# [ 1, 2, 3, 4, 5 ],
# [ 1, 3, 7, 13, 21 ]
# N = (d^2  d + 1)
# = ($d**2  $d + 1)
# = (($d  1)*$d + 1)
# d = 1/2 + sqrt(1 * $n + 3/4)
# = (1 + sqrt(4*$n  3)) / 2
#
# wider=3
# [ 2, 3, 4, 5 ],
# [ 6, 13, 22, 33 ]
# N = (d^2 + 2 d  2)
# = ($d**2 + 2*$d  2)
# = (($d + 2)*$d  2)
# d = 1 + sqrt(1 * $n + 3)
#
# wider=5
# [ 2, 3, 4, 5 ],
# [ 8, 17, 28, 41 ]
# N = (d^2 + 4 d  4)
# = ($d**2 + 4*$d  4)
# = (($d + 4)*$d  4)
# d = 2 + sqrt(1 * $n + 8)
#
# wider=7
# [ 2, 3, 4, 5 ],
# [ 10, 21, 34, 49 ]
# N = (d^2 + 6 d  6)
# = ($d**2 + 6*$d  6)
# = (($d + 6)*$d  6)
# d = 3 + sqrt(1 * $n + 15)
#
#
# N = (d^2 + (w1)*d + 1w)
# d = (1w)/2 + sqrt($n + (w^2 + 2w  3)/4)
# = (1w + sqrt(4*$n + (w3)(w+1))) / 2
#
# extra subtract d+w1
# Nbase = (d^2 + (w1)*d + 1w) + d+w1
# = d^2 + w*d
sub n_to_dxdy {
my ($self, $n) = @_;
### n_to_dxdy(): $n
$n = $n  $self>{'n_start'}; # starting $n==0, warn if $n==undef
if ($n < 0) {
#### before n_start ...
return;
}
my $w = $self>{'wider'};
my $d = int((1$w + sqrt(int(4*$n) + ($w+2)*$w+1)) / 2);
my $int = int($n);
$n = $int; # fraction 0 <= $n < 1
$int = ($d+$w)*$d1;
### $d
### $w
### $n
### $int
my ($dx, $dy);
if ($int <= 0) {
if ($int < 0) {
### horizontal ...
$dx = 1;
$dy = 0;
} else {
### corner horiz to vert ...
$dx = 1$n;
$dy = $n;
}
} else {
if ($int < $d) {
### vertical ...
$dx = 0;
$dy = 1;
} else {
### corner vert to horiz ...
$dx = $n;
$dy = 1$n;
}
}
unless ($d % 2) {
### rotate +180 for even d ...
$dx = $dx;
$dy = $dy;
}
### result: "$dx, $dy"
return ($dx,$dy);
}
# old bit:
#
# wider==0
# base from twoway diagonal topright and bottomleft
# s even for topright diagonal doing top leftwards then left downwards
# s odd for bottomleft diagonal doing bottom rightwards then right pupwards
# s = [ 0, 1, 2, 3, 4, 5, 6 ]
# N = [ 1, 1, 3, 7, 13, 21, 31 ]
# +0 +2 +4 +6 +8 +10
# 2 2 2 2 2
#
# n = (($d  1)*$d + 1)
# s = 1/2 + sqrt(1 * $n + 3/4)
# = .5 + sqrt ($n  .75)
#
#
#
sub _NOTDOCUMENTED_n_to_figure_boundary {
my ($self, $n) = @_;
### _NOTDOCUMENTED_n_to_figure_boundary(): $n
# adjust to N=1 at origin X=0,Y=0
$n = $n  $self>{'n_start'} + 1;
if ($n < 1) {
return undef;
}
my $wider = $self>{'wider'};
if ($n <= $wider) {
# single block row
# ++++
#  1  ...  $n  boundary = 2*N + 2
# ++++
return 2*$n + 2;
}
my $d = int((sqrt(int(4*$n) + $wider*$wider  2)  $wider) / 2);
### $d
### $wider
### cmp: $d*($d+1+$wider) + $wider + 1
if ($n > $d*($d+1+$wider)) {
$wider++;
### increment for +2 after turn ...
}
return 4*$d + 2*$wider + 2;
}
#
1;
__END__
=for stopwords Stanislaw Ulam pronic PlanePath Ryde MathPlanePath Ulam's Honaker's decagonal OEIS Nbase sqrt BigRat Nrem wl wr Nsig incrementing
=head1 NAME
Math::PlanePath::SquareSpiral  integer points drawn around a square (or rectangle)
=head1 SYNOPSIS
use Math::PlanePath::SquareSpiral;
my $path = Math::PlanePath::SquareSpiral>new;
my ($x, $y) = $path>n_to_xy (123);
=head1 DESCRIPTION
This path makes a square spiral,
=cut
# mathimage path=SquareSpiral all output=numbers_dash size=40x16
=pod
37363534333231 3
 
38 1716151413 30 2
   
39 18 543 12 29 1
     
40 19 6 12 11 28 ... < Y=0
     
41 20 78910 27 52 1
   
42 212223242526 51 2
 
4344454647484950 3
^
3 2 1 X=0 1 2 3 4
See F in the sources for a simple program
printing these numbers.
This path is well known from Stanislaw Ulam finding interesting straight
lines when plotting the prime numbers on it. The cover of Scientific
American March 1964 featured this spiral,
=over
L
L
=back
See F in the sources for a standalone program,
or see L using this C to draw this pattern and
more.
=head2 Straight Lines
XThe perfect squares 1,4,9,16,25 fall on two diagonals with
the even perfect squares going to the upper left and the odd squares to the
lower right. The Xpronic numbers 2,6,12,20,30,42 etc k^2+k
half way between the squares fall on similar diagonals to the upper right
and lower left. The decagonal numbers 10,27,52,85 etc 4*k^23*k go
horizontally to the right at Y=1.
In general straight lines and diagonals are 4*k^2 + b*k + c. b=0 is the
even perfect squares up to the left, then incrementing b is an eighth turn
anticlockwise, or clockwise if negative. So b=1 is horizontal West, b=2
diagonally down SouthWest, b=3 down South, etc.
Honaker's primegenerating polynomial 4*k^2 + 4*k + 59 goes down to the
right, after the first 30 or so values loop around a bit.
=head2 Wider
An optional C parameter makes the path wider, becoming a rectangle
spiral instead of a square. For example
$path = Math::PlanePath::SquareSpiral>new (wider => 3);
gives
2928272625242322 2
 
30 1110 9 8 7 6 21 1
   
31 12 1 2 3 4 5 20 < Y=0
  
32 13141516171819 1

33343536... 2
^
4 3 2 1 X=0 1 2 3
The centre horizontal 1 to 2 is extended by C many further places,
then the path loops around that shape. The starting point 1 is shifted to
the left by ceil(wider/2) places to keep the spiral centred on the origin
X=0,Y=0.
Widening doesn't change the nature of the straight lines which arise, it
just rotates them around. For example in this wider=3 example the perfect
squares are still on diagonals, but the even squares go towards the bottom
left (instead of top left when wider=0) and the odd squares to the top right
(instead of the bottom right).
Each loop is still 8 longer than the previous, as the widening is basically
a constant amount in each loop.
=head2 N Start
The default is to number points starting N=1 as shown above. An optional
C can give a different start with the same shape. For example to
start at 0,
=cut
# mathimage path=SquareSpiral,n_start=0 all output=numbers_dash size=35x16
=pod
n_start => 0
1615141312 ...
  
17 432 11 28
    
18 5 01 10 27
   
19 6789 26
 
202122232425
The only effect is to push the N values around by a constant amount. It
might help match coordinates with something else zerobased.
=head2 Corners
Other spirals can be formed by cutting the corners of the square so as to go
around faster. See the following modules,
Corners Cut Class
 
1 HeptSpiralSkewed
2 HexSpiralSkewed
3 PentSpiralSkewed
4 DiamondSpiral
The C is a reshaped C looping at the same
rate. It shifts corners but doesn't cut them.
=head1 FUNCTIONS
See L for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::SquareSpiralEnew ()>
=item C<$path = Math::PlanePath::SquareSpiralEnew (wider =E $integer, n_start =E $n)>
Create and return a new square spiral object. An optional C
parameter widens the spiral path, it defaults to 0 which is no widening.
=item C<($x,$y) = $pathEn_to_xy ($n)>
Return the X,Y coordinates of point number C<$n> on the path.
For C<$n E 1> the return is an empty list, as the path starts at 1.
=item C<$n = $pathExy_to_n ($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 N
in the path as centred in a square of side 1, so the entire plane is
covered.
=back
=head1 FORMULAS
=head2 N to X,Y
There's a few ways to break an N into a side and offset into the side. One
convenient way is to treat a loop as starting at the bottom right corner, so
N=2,10,26,50,etc, If the first at N=2 is reckoned loop number d=1 then
Nbase = 4*d^2  4*d + 2
For example d=3 is Nbase=4*3^24*3+2=26 at X=3,Y=2. The biggest d with
Nbase E= N can be found by inverting with the usual quadratic formula
d = floor (1/2 + sqrt(N/4  1/4))
For Perl it's good to keep the sqrt argument an integer (when a UV integer
is bigger than an NV float, and for BigRat accuracy), so rearranging
d = floor ((1+sqrt(N1)) / 2)
So Nbase from this d leaves a remainder which is an offset into the loop
Nrem = N  Nbase
= N  (4*d^2  4*d + 2)
The loop starts at X=d,Y=d1 and has sides length 2d, 2d+1, 2d+1 and 2d+2,
2d
++ < Y=d
 
2d   2d1
 . 
 
 + X=d,Y=d+1

++ < Y=d
2d+1
^
X=d
The X,Y for an Nrem is then
side Nrem range X,Y result
  
right Nrem <= 2d1 X = d
Y = d+1+Nrem
top 2d1 <= Nrem <= 4d1 X = d(Nrem(2d1)) = 3d1Nrem
Y = d
left 4d1 <= Nrem <= 6d1 X = d
Y = d(Nrem(4d1)) = 5d1Nrem
bottom 6d1 <= Nrem X = d+(Nrem(6d1)) = 7d+1+Nrem
Y = d
The corners Nrem=2d1, Nrem=4d1 and Nrem=6d1 get the same result from the
two sides that meet so it doesn't matter if the high comparison is "E"
or "E=".
The bottom edge runs through to Nrem E 8d, but there's no need to
check that since d=floor(sqrt()) above ensures Nrem is within the loop.
A small simplification can be had by subtracting an extra 4d1 from Nrem to
make negatives for the right and top sides and positives for the left and
bottom.
Nsig = N  Nbase  (4d1)
= N  (4*d^2  4*d + 2)  (4d1)
= N  (4*d^2 + 1)
side Nsig range X,Y result
  
right Nsig <= 2d X = d
Y = d+(Nsig+2d) = 3d+Nsig
top 2d <= Nsig <= 0 X = dNsig
Y = d
left 0 <= Nsig <= 2d X = d
Y = dNsig
bottom 2d <= Nsig X = d+1+(Nsig(2d+1)) = Nsig3d
Y = d
=head2 N to X,Y with Wider
With the C parameter stretching the spiral loops the formulas above
become
Nbase = 4*d^2 + (4+2w)*d + 2w
d = floor ((2w + sqrt(4N + w^2  4)) / 4)
Notice for Nbase the w is a term 2*w*d, being an extra 2*w for each loop.
The left offset ceil(w/2) described above (L) for the N=1 starting
position is written here as wl, and the other half wr arises too,
wl = ceil(w/2)
wr = floor(w/2) = w  wl
The horizontal lengths increase by w, and positions shift by wl or wr, but
the verticals are unchanged.
2d+w
++ < Y=d
 
2d   2d1
 . 
 
 + X=d+wr,Y=d+1

++ < Y=d
2d+1+w
^
X=dwl
The Nsig formulas then have w, wl or wr variously inserted. In all cases if
w=wl=wr=0 then they simplify to the plain versions.
Nsig = N  Nbase  (4d1+w)
= N  ((4d + 2w)*d + 1)
side Nsig range X,Y result
  
right Nsig <= (2d+w) X = d+wr
Y = d+(Nsig+2d+w) = 3d+w+Nsig
top (2d+w) <= Nsig <= 0 X = dwlNsig
Y = d
left 0 <= Nsig <= 2d X = dwl
Y = dNsig
bottom 2d <= Nsig X = d+1wl+(Nsig(2d+1)) = Nsigwl3d
Y = d
=head2 Rectangle to N Range
Within each row the minimum N is on the X=Y diagonal and N values increases
monotonically as X moves away to the left or right. Similarly in each
column there's a minimum N on the X=Y opposite diagonal, or X=Y+1 diagonal
when X negative, and N increases monotonically as Y moves away from there up
or down. When widerE0 the location of the minimum changes, but N is
still monotonic moving away from the minimum.
On that basis the maximum N in a rectangle is at one of the four corners,

x1,y2 MM x2,y2 corner candidates
   for maximum N
O
  
  
x1,y1 MM x1,y1

=head1 OEIS
This path is in Sloane's Online Encyclopedia of Integer Sequences in various
forms. Summary at
=over
L
=back
And various sequences,
=over
L (etc),
L
=back
wider=0 (the default)
A174344 X coordinate
A214526 abs(X)+abs(Y) "Manhattan" distance
A079813 abs(dY), being k 0s followed by k 1s
A063826 direction 1=right,2=up,3=left,4=down
A027709 boundary length of N unit squares
A078633 grid sticks to make N unit squares
A033638 N turn positions (extra initial 1, 1)
A172979 N turn positions which are primes too
A054552 N values on X axis (East)
A054556 N values on Y axis (North)
A054567 N values on negative X axis (West)
A033951 N values on negative Y axis (South)
A054554 N values on X=Y diagonal (NE)
A054569 N values on negative X=Y diagonal (SW)
A053755 N values on X=Y opp diagonal X<=0 (NW)
A016754 N values on X=Y opp diagonal X>=0 (SE)
A200975 N values on all four diagonals
A137928 N values on X=Y+1 opposite diagonal
A002061 N values on X=Y diagonal pos and neg
A016814 (4k+1)^2, every second N on southeast diagonal
A143856 N values on ENE slope dX=2,dY=1
A143861 N values on NNE slope dX=1,dY=2
A215470 N prime and >=4 primes among its 8 neighbours
A214664 X coordinate of prime N (Ulam's spiral)
A214665 Y coordinate of prime N (Ulam's spiral)
A214666 X \ reckoning spiral starting West
A214667 Y /
A053999 prime[N] on X=Y opp diagonal X>=0 (SE)
A054551 prime[N] on the X axis (E)
A054553 prime[N] on the X=Y diagonal (NE)
A054555 prime[N] on the Y axis (N)
A054564 prime[N] on X=Y opp diagonal X<=0 (NW)
A054566 prime[N] on negative X axis (W)
A090925 permutation N at rotate +90
A090928 permutation N at rotate +180
A090929 permutation N at rotate +270
A090930 permutation N at clockwise spiralling
A020703 permutation N at rotate +90 and go clockwise
A090861 permutation N at rotate +180 and go clockwise
A090915 permutation N at rotate +270 and go clockwise
A185413 permutation N at 1X,Y
being rotate +180, offset X+1, clockwise
A068225 permutation N to the N to its right, X+1,Y
A121496 run lengths of consecutive N in that permutation
A068226 permutation N to the N to its left, X1,Y
A020703 permutation N at transpose Y,X
(clockwise <> anticlockwise)
A033952 digits on negative Y axis
A033953 digits on negative Y axis, starting 0
A033988 digits on negative X axis, starting 0
A033989 digits on Y axis, starting 0
A033990 digits on X axis, starting 0
A062410 total sum previous row or column
wider=1
A069894 N on SouthWest diagonal
The following have "offset 0" in the OEIS and therefore are based on
starting from N=0.
n_start=0
A180714 X+Y coordinate sum
A053615 abs(XY), runs n to 0 to n, distance to nearest pronic
A001107 N on X axis
A033991 N on Y axis
A033954 N on negative Y axis, second 10gonals
A002939 N on X=Y diagonal NorthEast
A016742 N on NorthWest diagonal, 4*k^2
A002943 N on SouthWest diagonal
A156859 N on Y axis positive and negative
=head1 SEE ALSO
L,
L
L,
L,
L,
L
L
L
X11 cursor font "box spiral" cursor which is this style (but going
clockwise).
=head1 HOME PAGE
L
=head1 LICENSE
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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
MathPlanePath. If not, see .
=cut