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Math-NumSeq-58
River stage zero No dependents
3 ++
/ Math::NumSeq::PisanoPeriodSteps

NAME

Math::NumSeq::PisanoPeriodSteps -- Fibonacci frequency and Leonardo logarithm

SYNOPSIS

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

DESCRIPTION

This is the number of times the PisanoPeriod must be applied before reaching an unchanging value.

    # starting i=1
    0, 4, 3, 2, 3, 1, 2, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 1, 2, ...

As per Fulton and Morris

    "On arithmetical functions related to the Fibonacci numbers",
    Acta Arithmetica, volume 16, 1969, pages 105-110.
    http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1621.pdf

repeatedly applying the PisanoPeriod eventually reaches an m which is unchanging, ie. for which PisanoPeriod(m)==m. For example i=5 goes

    PisanoPeriod(5)=20
    PisanoPeriod(20)=60
    PisanoPeriod(60)=60
    PisanoPeriod(120)=120
    so value=3 applications until to reach unchanging 120

Leonardo Logarithm

The unchanging period reached is always of the form

    m = 24 * 5^(l-1)

The "l" exponent is the Leonardo logarithm. Option values_type => "log" returns that as the sequence values.

    # starting i=1
    0, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, ...

For example the i=5 above ends at m=120=24*5^1 so l-1=1 is l=2 for the sequence value.

FUNCTIONS

See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::PisanoPeriodSteps->new ()
$seq = Math::NumSeq::PisanoPeriodSteps->new (values_type => $str)

Create and return a new sequence object.

Random Access

$value = $seq->ith($i)

Return the count or logarithm of $i.

SEE ALSO

Math::NumSeq, Math::NumSeq::Fibonacci

HOME PAGE

http://user42.tuxfamily.org/math-numseq/index.html

LICENSE

Copyright 2012, 2013 Kevin Ryde

Math-NumSeq 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-NumSeq 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-NumSeq. If not, see <http://www.gnu.org/licenses/>.