# Copyright 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 .
# Classic Sequences
# http://oeis.org/classic.html
#
# Clark Kimberling
# http://faculty.evansville.edu/ck6/integer/intersp.html
#
# cf A175004 similar to wythoff but rows recurrence
# r(n1)+r(n2)+1 extra +1 in each step
# floor(n*phi+2/phi)
#
# cf Stolarsky round_nearest(n*phi)
# A035506 stolarsky by diagonals
# A035507 inverse
# A007067 stolarsky first column
# Maybe:
# my ($x,$y) = $path>pair_to_xy($a,$b);
# Return the $x,$y which has ($a,$b).
# Advance $a,$b if before start of row.
# Carlitz and Hoggatt "Fibonacci Representations", Fibonacci Quarterly,
# volume 10, number 1, January 1972
# http://www.fq.math.ca/101.html
# http://www.fq.math.ca/Scanned/101/carlitz1.pdf
package Math::PlanePath::WythoffArray;
use 5.004;
use strict;
#use List::Util 'max';
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 116;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::Base::Digits
'bit_split_lowtohigh';
# uncomment this to run the ### lines
#use Smart::Comments;
use constant parameter_info_array =>
[ { name => 'x_start',
display => 'X start',
type => 'integer',
default => 0,
width => 3,
description => 'Starting X coordinate.',
},
{ name => 'y_start',
display => 'Y start',
type => 'integer',
default => 0,
width => 3,
description => 'Starting Y coordinate.',
},
];
use constant default_n_start => 1;
use constant class_x_negative => 0;
use constant class_y_negative => 0;
sub x_minimum {
my ($self) = @_;
return $self>{'x_start'};
}
sub y_minimum {
my ($self) = @_;
return $self>{'y_start'};
}
use constant absdx_minimum => 1;
use constant dir_maximum_dxdy => (3,1); # N=4 to N=5 dX=3,dY=1
#
sub new {
my $self = shift>SUPER::new(@_);
$self>{'x_start'} = 0;
$self>{'y_start'} = 0;
return $self;
}
sub xy_is_visited {
my ($self, $x, $y) = @_;
return ((round_nearest($x) >= $self>{'x_start'})
&& (round_nearest($y) >= $self>{'y_start'}));
}
#
# 4  12 20 32 52 84 136 220 356 576 932 1508
# 3  9 15 24 39 63 102 165 267 432 699 1131
# 2  6 10 16 26 42 68 110 178 288 466 754
# 1  4 7 11 18 29 47 76 123 199 322 521
# Y=0  1 2 3 5 8 13 21 34 55 89 144
# +
# X=0 1 2 3 4 5 6 7 8 9 10
# 13,8,5,3,2,1
# 4 = 3+1 > 1
# 6 = 5+1 > 2
# 9 = 8+1 > 3
# 12 = 8+3+1 > 3+1=4
# 14 = 13+1 > 5
sub n_to_xy {
my ($self, $n) = @_;
### WythoffArray n_to_xy(): $n
if ($n < 1) { return; }
if (is_infinite($n)  $n == 0) { return ($n,$n); }
{
# fractions on straight line between integer points
my $int = int($n);
if ($n != $int) {
my $frac = $n  $int; # inherit possible BigFloat/BigRat
my ($x1,$y1) = $self>n_to_xy($int);
my ($x2,$y2) = $self>n_to_xy($int+1);
my $dx = $x2$x1;
my $dy = $y2$y1;
return ($frac*$dx + $x1, $frac*$dy + $y1);
}
$n = $int;
}
# f1+f0 > i
# f0 > if1
# check if1 as the stopping point, so that if i=UV_MAX then won't
# overflow a UV trying to get to f1>=i
#
my @fibs;
{
my $f0 = ($n * 0); # inherit bignum 0
my $f1 = $f0 + 1; # inherit bignum 1
while ($f0 <= $n$f1) {
($f1,$f0) = ($f1+$f0,$f1);
push @fibs, $f1; # starting $fibs[0]=1
}
}
### @fibs
# indices into fib[] which are the Fibonaccis adding up to $n
my @indices;
for (my $i = $#fibs; $i >= 0; $i) {
### at: "n=$n f=".$fibs[$i]
if ($n >= $fibs[$i]) {
push @indices, $i;
$n = $fibs[$i];
### sub: "$fibs[$i] to n=$n"
$i;
}
}
### @indices
# X is low index, ie. how many low 0 bits in Zeckendorf form
my $x = pop @indices;
### $x
# Y is indices shifted down by $x and 2 more
my $y = 0;
my $shift = $x+2;
foreach my $i (@indices) {
### y add: "ishift=".($i$shift)." fib=".$fibs[$i$shift]
$y += $fibs[$i$shift];
}
### $shift
### $y
return ($x+$self>{'x_start'},$y+$self>{'y_start'});
}
# phi = (sqrt(5)+1)/2
# (y+1)*phi = (y+1)*(sqrt(5)+1)/2
# = ((y+1)*sqrt(5)+(y+1))/2
# = (sqrt(5*(y+1)^2)+(y+1))/2
#
# from x=0,y=0
# N = floor((y+1)*Phi) * Fib(x+2) + y*Fib(x+1)
#
sub xy_to_n {
my ($self, $x, $y) = @_;
### WythoffArray xy_to_n(): "$x, $y"
$x = round_nearest($x)  $self>{'x_start'};
$y = round_nearest($y)  $self>{'y_start'};
if ($x < 0  $y < 0) {
return undef;
}
my $zero = $x * 0 * $y;
$x += 2;
if (is_infinite($x)) { return $x; }
if (is_infinite($y)) { return $y; }
my @bits = bit_split_lowtohigh($x);
### @bits
pop @bits; # discard high 1bit
my $yplus1 = $zero + $y+1; # inherit bigint from $x perhaps
# spectrum(Y+1) so Y,Ybefore are notional two values at X=2 and X=1
my $ybefore = int((sqrt(5*$yplus1*$yplus1) + $yplus1) / 2);
### $ybefore
# k=1, Fk1=F[k1]=0, Fk=F[k]=1
my $Fk1 = $zero;
my $Fk = 1 + $zero;
my $add = 2;
while (@bits) {
### remaining bits: @bits
### Fk1: "$Fk1"
### Fk: "$Fk"
# two squares and some adds
# F[2k+1] = 4*F[k]^2  F[k1]^2 + 2*(1)^k
# F[2k1] = F[k]^2 + F[k1]^2
# F[2k] = F[2k+1]  F[2k1]
#
$Fk *= $Fk;
$Fk1 *= $Fk1;
my $F2kplus1 = 4*$Fk  $Fk1 + $add;
$Fk1 += $Fk; # F[2k1]
my $F2k = $F2kplus1  $Fk1;
if (pop @bits) { # high to low
$Fk1 = $F2k; # F[2k]
$Fk = $F2kplus1; # F[2k+1]
$add = 2;
} else {
# $Fk1 is F[2k1] already
$Fk = $F2k; # F[2k]
$add = 2;
}
}
### final pair ...
### Fk1: "$Fk1"
### Fk: "$Fk"
### @bits
return ($Fk*$ybefore + $Fk1*$y);
}
# exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### WythoffArray rect_to_n_range(): "$x1,$y1 $x2,$y2"
$x1 = round_nearest($x1);
$y1 = round_nearest($y1);
$x2 = round_nearest($x2);
$y2 = round_nearest($y2);
($x1,$x2) = ($x2,$x1) if $x1 > $x2;
($y1,$y2) = ($y2,$y1) if $y1 > $y2;
if ($x2 < $self>{'x_start'}  $y2 < $self>{'y_start'}) {
### all outside first quadrant ...
return (1, 0);
}
# bottom left into first quadrant
$x1 = max($x1, $self>{'x_start'});
$y1 = max($y1, $self>{'y_start'});
return ($self>xy_to_n($x1,$y1), # bottom left
$self>xy_to_n($x2,$y2)); # top right
}
1;
__END__
=for stopwords eg Ryde ie MathPlanePath Wythoff Zeckendorf concecutive fibbinary bignum OEIS Stolarsky Morrison's Knott Generalising
=head1 NAME
Math::PlanePath::WythoffArray  table of Fibonacci recurrences
=head1 SYNOPSIS
use Math::PlanePath::WythoffArray;
my $path = Math::PlanePath::WythoffArray>new;
my ($x, $y) = $path>n_to_xy (123);
=head1 DESCRIPTION
XXThis path is the Wythoff array by David
R. Morrison
=over
"A Stolarsky Array of Wythoff Pairs", in Collection of Manuscripts Related
to the Fibonacci Sequence, pages 134 to 136, The Fibonacci Association,
1980. L
=back
It's an array of Fibonacci recurrences which positions each N according to
Zeckendorf base trailing zeros.
=cut
# mathimage path=WythoffArray output=numbers all size=60x16
=pod
15  40 65 105 170 275 445 720 1165 1885 3050 4935
14  38 62 100 162 262 424 686 1110 1796 2906 4702
13  35 57 92 149 241 390 631 1021 1652 2673 4325
12  33 54 87 141 228 369 597 966 1563 2529 4092
11  30 49 79 128 207 335 542 877 1419 2296 3715
10  27 44 71 115 186 301 487 788 1275 2063 3338
9  25 41 66 107 173 280 453 733 1186 1919 3105
8  22 36 58 94 152 246 398 644 1042 1686 2728
7  19 31 50 81 131 212 343 555 898 1453 2351
6  17 28 45 73 118 191 309 500 809 1309 2118
5  14 23 37 60 97 157 254 411 665 1076 1741
4  12 20 32 52 84 136 220 356 576 932 1508
3  9 15 24 39 63 102 165 267 432 699 1131
2  6 10 16 26 42 68 110 178 288 466 754
1  4 7 11 18 29 47 76 123 199 322 521
Y=0  1 2 3 5 8 13 21 34 55 89 144
+
X=0 1 2 3 4 5 6 7 8 9 10
All rows have the Fibonacci style recurrence
W(X+1) = W(X) + W(X1)
eg. X=4,Y=2 is N=42=16+26, sum of the two values to its left
XX axis N=1,2,3,5,8,etc is the Fibonacci numbers.
XThe row Y=1 above them N=4,7,11,18,etc is the Lucas numbers.
XY axis N=1,4,6,9,12,etc is the "spectrum" of the golden
ratio, meaning its multiples rounded down to an integer.
phi = (sqrt(5)+1)/2
spectrum(k) = floor(phi*k)
N on Y axis = Y + spectrum(Y+1)
Eg. Y=5 N=5+floor((5+1)*phi)=14
The recurrence in each row starts as if the row was preceded by two values Y
and spectrum(Y+1) which can be thought of adding to be Y+spectrum(Y+1) on
the Y axis, then Y+2*spectrum(Y+1) in the X=1 column, etc.
If the first two values in a row have a common factor then that factor
remains in all subsequent sums. For example the Y=2 row starts with two
even numbers N=6,N=10 so all N values in the row are even.
Every N from 1 upwards occurs precisely once in the table. The recurrence
means that in each row N grows roughly as a power phi^X, the same as the
Fibonacci numbers. This means they become large quite quickly.
=head2 Zeckendorf Base
XThe N values are arranged according to trailing zero bits
when N is represented in the Zeckendorf base. The Zeckendorf base expresses
N as a sum of Fibonacci numbers, choosing at each stage the largest possible
Fibonacci. For example
Fibonacci numbers F[0]=1, F[1]=2, F[2]=3, F[3]=5, etc
45 = 34 + 8 + 3
= F[7] + F[4] + F[2]
= 10010100 1bits at 7,4,2
The Wythoff array written in Zeckendorf base bits is
=cut
# This table printed by tools/wythoffarrayzeck.pl
=pod
8  1000001 10000010 100000100 1000001000 10000010000
7  101001 1010010 10100100 101001000 1010010000
6  100101 1001010 10010100 100101000 1001010000
5  100001 1000010 10000100 100001000 1000010000
4  10101 101010 1010100 10101000 101010000
3  10001 100010 1000100 10001000 100010000
2  1001 10010 100100 1001000 10010000
1  101 1010 10100 101000 1010000
Y=0  1 10 100 1000 10000
+
X=0 1 2 3 4
The X coordinate is the number of trailing zeros, or equivalently the index
of the lowest Fibonacci used in the sum. For example in the X=3 column all
the N's there have F[3]=5 as their lowest term.
The Y coordinate is formed by removing the trailing "0100..00", ie. all
trailing zeros plus the "01" above them. For example,
N = 45 = Zeck 10010100
^^^^ strip low zeros and "01" above them
Y = Zeck(1001) = F[3]+F[0] = 5+1 = 6
The Zeckendorf form never has consecutive "11" bits, because after
subtracting an F[k] the remainder is smaller than the next lower F[k1].
Numbers with no concecutive "11" bits are sometimes called the fibbinary
numbers (see L).
Stripping low zeros is similar to what the C does with low zero
digits in an ordinary base such as binary (see
L). Doing it in the Zeckendorf base is like
taking out powers of the golden ratio phi=1.618.
=head2 Turn Sequence
The path turns
straight at N=2 and N=10
right N="...101" in Zeckendorf base
left otherwise
For example at N=12 the path turns to the right, since N=13 is on the right
hand side of the vector from N=11 to N=12. It's almost 180degrees around
and back, but on the right hand side.
4  12
3 
2 
1  11
Y=0  13
+
X=0 1 2 3 4 5
This happens because N=12 is Zeckendorf "10101" which ends "..101". For
such an ending N1 is "..100" and N+1 is "..1000". So N+1 has more trailing
zeros and hence bigger X smaller Y than N1 has. The way the curve grows in
a "concave" fashion means that therefore N+1 is on the righthand side.
 N N ending "..101"

 N+1 bigger X smaller Y
 N1 than N1
 N+1
+
Cases for N ending "..000", "..010" and "..100" can be worked through to see
that everything else turns left (or the initial N=2 and N=10 go straight
ahead).
On the Y axis all N values end "..01", with no trailing 0s. As noted above
stripping that "01" from N gives the Y coordinate. Those N ending "..101"
are therefore at Y coordinates which end "..1", meaning "odd" Y in
Zeckendorf base.
=head2 X,Y Start
Options C $x> and C $y> give a starting
position for the array. For example to start at X=1,Y=1
4  9 15 24 39 63 x_start => 1
3  6 10 16 26 42 y_start => 1
2  4 7 11 18 29
1  1 2 3 5 8
Y=0 
+
X=0 1 2 3 4 5
This can be helpful to work in rows and columns numbered from 1 instead of
from 0. Numbering from X=1,Y=1 corresponds to the array in Morrison's paper
above.
=head1 FUNCTIONS
See L for the behaviour common to all path
classes.
=over 4
=item C<$path = Math::PlanePath::WythoffArrayEnew ()>
=item C<$path = Math::PlanePath::WythoffArrayEnew (x_start =E $x, y_start =E $y)>
Create and return a new path object. The default C and C
are 0.
=item C<($x,$y) = $pathEn_to_xy ($n)>
Return the X,Y coordinates of point number C<$n> on the path. Points begin
at 1 and if C<$n E 1> then the return is an empty list.
=item C<$n = $pathExy_to_n ($x,$y)>
Return the N point number at coordinates C<$x,$y>. If C<$xE0> or
C<$yE0> (or the C or C options) then there's no N and
the return is C.
N values grow rapidly with C<$x>. Pass in a bignum type such as
C for full precision.
=item C<($n_lo, $n_hi) = $pathErect_to_n_range ($x1,$y1, $x2,$y2)>
The returned range is exact, meaning C<$n_lo> and C<$n_hi> are the smallest
and biggest in the rectangle.
=back
=head1 FORMULAS
=head2 Rectangle to N Range
Within each row increasing X is increasing N, and in each column increasing
Y is increasing N. So in any rectangle the minimum N is in the lower left
corner and the maximum N is in the upper right corner.
 N max
 +
  ^ 
   
  > 
 +
 N min
+
=head1 OEIS
The Wythoff array is in Sloane's Online Encyclopedia of Integer Sequences
in various forms,
=over
L (etc)
=back
x_start=0,y_start=0 (the defaults)
A035614 X, column numbered from 0
A191360 XY, the diagonal containing N
A019586 Y, the row containing N
A083398 max diagonal X+Y+1 for points 1 to N
x_start=1,y_start=1
A035612 X, column numbered from 1
A003603 Y, vertical parabudding sequence
A143299 Zeckendorf bit count in row Y
A185735 leftjustified row addition
A186007 row subtraction
A173028 row multiples
A173027 row of n * Fibonacci numbers
A220249 row of n * Lucas numbers
A003622 N on Y axis, odd Zeckendorfs "..1"
A020941 N on X=Y diagonal
A139764 N dropped down to X axis, ie. N value on the X axis,
being lowest Fibonacci used in the Zeckendorf form
A000045 N on X axis, Fibonacci numbers skipping initial 0,1
A000204 N on Y=1 row, Lucas numbers skipping initial 1,3
A001950 N+1 of N on Y axis, antispectrum of phi
A022342 N not on Y axis, even Zeckendorfs "..0"
A000201 N+1 of N not on Y axis, spectrum of phi
A003849 bool 1,0 if N on Y axis or not, being the Fibonacci word
A035336 N in second column
A160997 total N along antidiagonals X+Y=k
A188436 turn 1=right,0=left or straight, skip initial five 0s
A134860 N positions of right turns, Zeckendorf "..101"
A003622 Y coordinate of right turns, Zeckendorf "..1"
A114579 permutation N at transpose Y,X
A083412 permutation N by Diagonals from Y axis downwards
A035513 permutation N by Diagonals from X axis upwards
A064274 inverse permutation
=head1 SEE ALSO
L,
L,
L
L,
L,
L,
L,
L
Ron Knott, "Generalising the Fibonacci Series",
L
OEIS Classic Sequences, "The Wythoff Array and The ParaFibonacci Sequence",
L
=head1 HOME PAGE
L
=head1 LICENSE
Copyright 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