/ 
Math-NumSeq-55
River stage zero No dependents
/ Math::NumSeq::FibonacciWord

NAME

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

SYNOPSIS

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

DESCRIPTION

This is a sequence of 0s and 1s formed from the Fibonacci numbers.

starting i=0
0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, ...

The initial values are 0,1 then Fibonacci number F(k) many values are copied from the start to extend, so

0,1                                         initial
0,1,0                                       append 1 value
0,1,0,0,1                                   append 2 values
0,1,0,0,1,0,1,0                             append 3 values
0,1,0,0,1,0,1,0,0,1,0,0,1                   append 5 values
0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0   append 8 values
etc

The same sequence is had by starting with 0 and then repeatedly expanding

0 -> 0,1
1 -> 0

The result is also the Fibbinary numbers modulo 2. This can be seen most easily from the Zeckendorf base interpretation of those numbers since the base breakdown there works backwards from the above expansion subtracting Fibonacci numbers F(k) until reaching 0 or 1. (See "Zeckendorf Base" in Math::NumSeq::Fibbinary.)

Dense Fibonacci Word

Option fibonacci_word_type => "dense" selects the dense Fibonacci word

1,0,2,2,1,0,2,2,1,1,0,2,1,1,...

This is the above plain word taken two values at a time encoded as

plain    dense
 0,0       0    
 0,1       1
 1,0       2

For example the plain Fibonacci word starts 0,1 so the dense form starts 1. Adjacent 1,1 never occurs in the plain Fibonacci word, so there's no value 3 in the dense form.

FUNCTIONS

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

$seq = Math::NumSeq::FibonacciWord->new ()
$seq = Math::NumSeq::FibonacciWord->new (fibonacci_word_type => "dense")

Create and return a new sequence object.

Random Access

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

Return the $i'th value in the sequence. The first value is at i=0.

$bool = $seq->pred($value)

Return true if $value occurs in the sequence, which simply means 0 or 1, or for the dense Fibonacci word 0, 1 or 2.

SEE ALSO

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

Math::PlanePath::FibonacciWordFractal

HOME PAGE

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

LICENSE

Copyright 2011, 2012 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/>.