This file is indexed.

/usr/share/perl5/Math/PlanePath/PentSpiral.pm is in libmath-planepath-perl 117-1.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
# Copyright 2010, 2011, 2012, 2013, 2014 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/>.


package Math::PlanePath::PentSpiral;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;

use vars '$VERSION', '@ISA';
$VERSION = 117;
use Math::PlanePath;
@ISA = ('Math::PlanePath');

use Math::PlanePath::Base::Generic
  'round_nearest';

# uncomment this to run the ### lines
#use Smart::Comments;


use constant parameter_info_array =>
  [
   Math::PlanePath::Base::Generic::parameter_info_nstart1(),
  ];

sub x_negative_at_n {
  my ($self) = @_;
  return $self->n_start + 3;
}
sub y_negative_at_n {
  my ($self) = @_;
  return $self->n_start + 4;
}
sub _UNDOCUMENTED__dxdy_list_at_n {
  my ($self) = @_;
  return $self->n_start + 6;
}

use constant dx_minimum => -2;
use constant dx_maximum => 2;
use constant dy_minimum => -1;
use constant dy_maximum => 1;
use constant _UNDOCUMENTED__dxdy_list => (2,0,   # E by 2
                           1,1,   # NE
                           -2,1,  # WNW
                           -2,-1, # WSW
                           1,-1,  # SE
                          );
use constant absdx_minimum => 1;
use constant dsumxy_minimum => -3; # SW -2,-1
use constant dsumxy_maximum => 2;  # dX=+2 and NE diag
use constant ddiffxy_minimum => -3; # NW dX=-2,dY=+1
use constant ddiffxy_maximum => 2;
use constant dir_maximum_dxdy => (1,-1); # South-East

#------------------------------------------------------------------------------

sub new {
  my $self = shift->SUPER::new(@_);
  if (! defined $self->{'n_start'}) {
    $self->{'n_start'} = $self->default_n_start;
  }
  return $self;
}

# base South-West diagonal
#   d = [  1,  2,  3,  4 ]
#   n = [  0,  4, 13, 27 ]
# N = (5/2 d^2 - 7/2 d + 1)
#   = (5/2*$d**2 - 7/2*$d + 1)
#   = ((5/2*$d - 7/2)*$d + 1)
# d = 7/10 + sqrt(2/5 * $n + 9/100)
#   = (sqrt(40*$n + 9) + 7) / 10
#
# split Y axis
#   d = [  1,  2,  3 ]
#   n = [  2,  9, 21 ]
# N = ((5/2*$d - 1/2)*$d)

sub n_to_xy {
  my ($self, $n) = @_;
  #### n_to_xy: $n

  # adjust to N=0 at origin X=0,Y=0
  $n = $n - $self->{'n_start'};
  if ($n < 0) { return; }

  my $d = int( (sqrt(40*$n+9)+7) / 10);
  $n -= (5*$d-1)*$d/2;

  if ($n < -$d) {
    $n += 2*$d;
    if ($n < 1) {
      # bottom horizontal
      return (2*$n+$d-1, -$d+1);
    } else {
      # lower right diagonal ...
      return ($n+$d, $n-$d);
    }
  } else {
    if ($n <= $d) {
      ### top 2,1 slope left and right diagonals ...
      return (-2*$n,
              -abs($n) + $d);
    } else {
      ### lower left diagonal ...
      return ($n - 3*$d,
              -$n + $d);
    }
  }
}

sub xy_to_n {
  my ($self, $x, $y) = @_;

  $x = round_nearest ($x);
  $y = round_nearest ($y);

  # nothing on odd points
  # when y>=0 any odd x is not covered
  # when y<0 the uncovered alternates, x even on y=-1, x odd on y=-2, x even
  # y=-3 etc
  if (($x%2) ^ ($y < 0 ? $y%2 : 0)) {
    return undef;
  }

  if ($y >= 0) {
    ### top left and right slopes
    # vertical at x=0
    #   d = [ 1, 2, 3 ]
    #   n = [ 3, 10, 22 ]
    #   n = (5/2*$d**2 + -1/2*$d + 1)
    #
    ### assert: ($x%2)==0
    $x /= 2;
    my $d = abs($x) + $y;
    return (5*$d - 1)*$d/2 - $x + $self->{'n_start'};
  }

  if ($x < $y) {
    ### lower left slope
    # horizontal leftwards at y=0
    #   d = [ 1,  2,  3 ]
    #   n = [ 4, 12, 25 ]
    #   n = (5/2*$d**2 + 1/2*$d + 1)
    #     = (2.5*$d + 0.5)*$d + 1
    my $d = -($x+$y)/2;
    return (5*$d + 1)*$d/2 - $y + $self->{'n_start'};
  }

  if ($x > -$y) {
    ### lower right slope
    # horizontal rightwards at y=0
    #   d = [ 1, 2, 3, ]
    #   n = [ 2, 8, 19,]
    #   n = (5/2*$d**2 + -3/2*$d + 1)
    #     = (2.5*$d - 1.5)*$d + 1
    my $d = ($x-$y)/2;
    return (5*$d - 3)*$d/2 + $y + $self->{'n_start'};
  }

  ### bottom horizontal
  # vertical downwards at x=0 is
  #   y = [  -1, -2,   -3 ]
  #   n = [ 5.5, 15, 29.5 ]
  #   n = (5/2*$y**2 + -2*$y + 1)
  #     = (2.5*$y - 2)*$y + 1
  # so
  #   N = (2.5*$y - 2)*$y + 1  +  $x/2
  #     = ((5*$y - 4)*$y + $x)/2 + 1
  #
  return ((5*$y-4)*$y + $x)/2 + $self->{'n_start'};
}

# not exact
sub rect_to_n_range {
  my ($self, $x1,$y1, $x2,$y2) = @_;
  ### PentSpiral rect_to_n_range(): $x1,$y1, $x2,$y2

  my $d = 0;
  foreach my $x ($x1, $x2) {
    $x = round_nearest ($x);
    foreach my $y ($y1, $y2) {
      $y = round_nearest ($y);

      my $this_d = 1 + ($y >= 0     ? abs($x) + $y
                        : $x < $y   ? -($x+$y)/2
                        : $x > -$y  ? ($x-$y)/2
                        : -$y);
      ### $x
      ### $y
      ### $this_d
      $d = max($d, $this_d);
    }
  }
  ### $d
  return ($self->{'n_start'},
          $self->{'n_start'} + 5*$d*($d-1)/2 + 2);
}

1;
__END__

=for stopwords Ryde Math-PlanePath OEIS

=head1 NAME

Math::PlanePath::PentSpiral -- integer points in a pentagonal shape

=head1 SYNOPSIS

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

=head1 DESCRIPTION

This path makes a pentagonal (five-sided) spiral with points spread out to
fit on a square grid.

                      22                              3

                23    10    21                        2

          24    11     3     9    20                  1

    25    12     4     1     2     8    19       <- Y=0

       26    13     5     6     7    18    ...       -1

          27    14    15    16    17    33           -2

             28    29    30    31    32              -2


     ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
    -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7

Each horizontal gap is 2, so for instance n=1 is at x=0,y=0 then n=2 is at
x=2,y=0.  The lower diagonals are 1 across and 1 down, so n=17 is at
x=4,y=-2 and n=18 is x=5,y=-1.  But the upper angles go 2 across and 1 up,
so n=20 is x=4,y=1 then n=21 is x=2,y=2.

The effect is to make the sides equal length, except for a kink at the lower
right corner.  Only every second square in the plane is used.  In the top
half (y>=0) those points line up, in the lower half (y<0) they're offset on
alternate rows.

=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=PentSpiral,n_start=0 --expression='i<=57?i:0' --output=numbers --size=120x11

=pod

    n_start => 0            38

                      39    21    37
                                           ...
                40    22     9    20    36    57

          41    23    10     2     8    19    35    56

    42    24    11     3     0     1     7    18    34    55

       43    25    12     4     5     6    17    33    54

          44    26    13    14    15    16    32    53

             45    27    28    29    30    31    52

                46    47    48    49    50    51

=head1 FUNCTIONS

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

=over 4

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

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

Create and return a new pentagon spiral object.

=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
point in the path as a square of side 1.

=back

=head1 FORMULAS

=head2 N to X,Y

It's convenient to work in terms of Nstart=0 and to take each loop as
beginning on the South-West diagonal,

                      21                loop d=3
                   --    --
                22          20
             --                --
          23                      19
       --                            --
    24                 0                18
      \                                /
       25          .                 17
         \                          /
          26    13----14----15----16
            \
             .

The SW diagonal is N=0,4,13,27,46,etc which is

    N = (5d-7)*d/2 + 1           # starting d=1 first loop

This can be inverted to get d from N

    d = floor( (sqrt(40*N + 9) + 7) / 10 )

Each side is length d, except the lower right diagonal slope which is d-1.
For the very first loop that lower right is length 0.

=head1 OEIS

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

=over

L<http://oeis.org/A140066> (etc)

=back

    n_start=1 (the default)
      A192136    N on X axis, (5*n^2 - 3*n + 2)/2
      A140066    N on Y axis
      A116668    N on X negative axis
      A005891    N on South-East diagonal, centred pentagonals
      A134238    N on South-West diagonal

    n_start=0
      A000566    N on X axis, heptagonal numbers
      A005476    N on Y axis
      A028895    N on South-East diagonal

=head1 SEE ALSO

L<Math::PlanePath>,
L<Math::PlanePath::PentSpiralSkewed>,
L<Math::PlanePath::HexSpiral>

=head1 HOME PAGE

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

=head1 LICENSE

Copyright 2010, 2011, 2012, 2013, 2014 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/>.

=cut