1Math::NumSeq::PisanoPerUisoedrStCeopnst(r3i)buted Perl DMoactuhm:e:nNtuamtSieoqn::PisanoPeriodSteps(3)
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NAME

6       Math::NumSeq::PisanoPeriodSteps -- Fibonacci frequency and Leonardo
7       logarithm
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SYNOPSIS

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

15       This is the number of times the "PisanoPeriod" must be applied before
16       reaching an unchanging value.
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18           0, 4, 3, 2, 3, 1, 2, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 1, 2, ...
19           starting i=1
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21       As per Fulton and Morris
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23           "On arithmetical functions related to the Fibonacci numbers", Acta
24           Arithmetica, volume 16, 1969, pages 105-110.
25           <http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1621.pdf>
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27       repeatedly applying the PisanoPeriod eventually reaches an m which is
28       unchanging, ie. for which PisanoPeriod(m)==m.  For example i=5 goes
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30           PisanoPeriod(5)=20
31           PisanoPeriod(20)=60
32           PisanoPeriod(60)=60
33           PisanoPeriod(120)=120
34           so value=3 applications until to reach unchanging 120
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36   Leonardo Logarithm
37       The unchanging period reached is always of the form
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39           m = 24 * 5^(l-1)
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41       The "l" exponent is the Leonardo logarithm.  Option "values_type =>
42       "log"" returns that as the sequence values.
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44           0, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, ...
45           starting i=1
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47       For example the i=5 above ends at m=120=24*5^1 so l-1=1 is l=2 for the
48       sequence value.
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FUNCTIONS

51       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
52       classes.
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54       "$seq = Math::NumSeq::PisanoPeriodSteps->new ()"
55       "$seq = Math::NumSeq::PisanoPeriodSteps->new (values_type => $str)"
56           Create and return a new sequence object.
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58   Random Access
59       "$value = $seq->ith($i)"
60           Return the count or logarithm of $i.
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SEE ALSO

63       Math::NumSeq, Math::NumSeq::Fibonacci
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HOME PAGE

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

69       Copyright 2012, 2013, 2014, 2016, 2019 Kevin Ryde
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71       Math-NumSeq is free software; you can redistribute it and/or modify it
72       under the terms of the GNU General Public License as published by the
73       Free Software Foundation; either version 3, or (at your option) any
74       later version.
75
76       Math-NumSeq is distributed in the hope that it will be useful, but
77       WITHOUT ANY WARRANTY; without even the implied warranty of
78       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
79       General Public License for more details.
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81       You should have received a copy of the GNU General Public License along
82       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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86perl v5.34.0                      2021-07-22Math::NumSeq::PisanoPeriodSteps(3)
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