Kevin Ryde > Math-NumSeq > Math::NumSeq::PisanoPeriodSteps

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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.

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

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.

    0, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, ...
    starting i=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, 2014 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/>.

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