1Math::NumSeq::FibonacciUWsoerrd(C3o)ntributed Perl DocumMeanttha:t:iNounmSeq::FibonacciWord(3)
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NAME

6       Math::NumSeq::FibonacciWord -- 0/1 related to Fibonacci numbers
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SYNOPSIS

9        use Math::NumSeq::FibonacciWord;
10        my $seq = Math::NumSeq::FibonacciWord->new;
11        my ($i, $value) = $seq->next;
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DESCRIPTION

14       This is a sequence of 0s and 1s formed from the Fibonacci numbers.
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16           0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, ...
17           starting i=0
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19       The initial values are 0,1.  Then Fibonacci number F(k) many values are
20       copied from the start to extend, repeatedly.
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22           0,1                                         initial
23           0,1,0                                       append 1 value
24           0,1,0,0,1                                   append 2 values
25           0,1,0,0,1,0,1,0                             append 3 values
26           0,1,0,0,1,0,1,0,0,1,0,0,1                   append 5 values
27           0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0   append 8 values
28           etc
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30   Morphism
31       The same sequence is had by starting with 0 and then repeatedly
32       expanding
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34           0 -> 0,1
35           1 -> 0
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37   Fibbinary and Zeckendorf
38       The result is also the Fibbinary numbers modulo 2, which is the least
39       significant bit of the Zeckendorf base representation of i.
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41       The Zeckendorf base breakdown subtracts Fibonacci numbers F(k) until
42       reaching 0 or 1.  This effectively undoes the above append expansion
43       procedure.  (See "Zeckendorf Base" in Math::NumSeq::Fibbinary.)
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45           start at i
46           until i=0 or i=1 do
47             subtract from i the largest Fibonacci number <= i
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49           final resulting i=0 or i=1 is Fibonacci word value
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51       For example i=11 has largest Fibonacci<=11 is 8, subtract that to leave
52       3.  From 3 the largest Fibonacci<=3 is 3 itself, subtract that to leave
53       0 which is the Fibonacci word value for i=11.
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55   Dense Fibonacci Word
56       Option "fibonacci_word_type => "dense"" selects the dense Fibonacci
57       word
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59           1,0,2,2,1,0,2,2,1,1,0,2,1,1,...
60           starting i=0
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62       This is the above plain word with each two values (not overlapping)
63       encoded in a binary style as
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65           plain pair   dense value
66           ----------   -----------
67               0,0           0
68               0,1           1
69               1,0           2
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71       For example the Fibonacci word starts 0,1 so the dense form starts 1.
72       A pair 1,1 never occurs in the plain Fibonacci word so there's no value
73       3 in the dense form.
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FUNCTIONS

76       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
77       classes.
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79       "$seq = Math::NumSeq::FibonacciWord->new ()"
80       "$seq = Math::NumSeq::FibonacciWord->new (fibonacci_word_type => $str)"
81           Create and return a new sequence object.  The "fibonacci_word_type"
82           option (a string) can be either
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84               "plain"   (the default)
85               "dense"
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87   Iterating
88       "$seq->seek_to_i($i)"
89           Move the current i so "next()" will return $i (and corresponding
90           value) on the next call.
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92   Random Access
93       "$value = $seq->ith($i)"
94           Return the $i'th value in the sequence.  The first value is at i=0.
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96       "$bool = $seq->pred($value)"
97           Return true if $value occurs in the sequence.  This simply means 0
98           or 1, or for the dense Fibonacci word 0, 1 or 2.
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SEE ALSO

101       Math::NumSeq, Math::NumSeq::Fibonacci, Math::NumSeq::Fibbinary
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103       Math::PlanePath::FibonacciWordFractal
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HOME PAGE

106       <http://user42.tuxfamily.org/math-numseq/index.html>
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LICENSE

109       Copyright 2011, 2012, 2013, 2014, 2016, 2019 Kevin Ryde
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111       Math-NumSeq is free software; you can redistribute it and/or modify it
112       under the terms of the GNU General Public License as published by the
113       Free Software Foundation; either version 3, or (at your option) any
114       later version.
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116       Math-NumSeq is distributed in the hope that it will be useful, but
117       WITHOUT ANY WARRANTY; without even the implied warranty of
118       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
119       General Public License for more details.
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121       You should have received a copy of the GNU General Public License along
122       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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126perl v5.32.1                      2021-01-27    Math::NumSeq::FibonacciWord(3)
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