Math::NumSeq::PisanoPeriod -- cycle length of Fibonacci numbers mod i
use Math::NumSeq::PisanoPeriod; my $seq = Math::NumSeq::PisanoPeriod->new; my ($i, $value) = $seq->next;
This is the length cycle of Fibonacci numbers modulo i.
1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, ... starting i=1
For example Fibonacci numbers modulo 4 repeat in a cycle of 6 numbers, so value=6.
Fibonacci 0, 1, 1, 2, 3, 5, 8,13,21,34,55,89,144,... mod 4 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0,... \--------------/ \--------------/ \--- repeating cycle of 6
The Fibonaccis are determined by a pair F[i],F[i+1] and there can be at most i*i many different pairs mod i, so there's always a finite repeating period. Since the Fibonaccis can go backwards as F[i-1]=F[i+1]-F[i] the modulo sequence is purely periodic, so the initial 0,1 is always part of the cycle.
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
$seq = Math::NumSeq::PisanoPeriod->new ()
Create and return a new sequence object.
$value = $seq->ith($i)
Return the Pisano period of $i
.
Math::NumSeq, Math::NumSeq::Fibonacci, Math::NumSeq::FibonacciWord
http://user42.tuxfamily.org/math-numseq/index.html
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.
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