NAME

Math::NumSeq::GolayRudinShapiro -- parity of adjacent 11 bit pairs

SYNOPSIS

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

DESCRIPTION

This is the Golay/Rudin/Shapiro sequence of +1 or -1 according as an even or odd number of adjacent 11 bit pairs in i.

GRS(i) = (-1) ^ (count 11 bit pairs)

starting from i=0
1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, ...

The first -1 is at i=3 which is binary 11 with a single 11 bit pair, then i=6 binary 110 likewise -1. Later for example i=14 is binary 1110 which has two adjacent 11 pairs (overlapping pairs count), so value=1.

The value is also the parity of the number of even-length runs of 1-bits in i. An even length run has an odd number of 11 pairs, so each of them is a -1 in the product. An odd-length run of 1-bits is an even number of 11 pairs so is +1 and has no effect on the result.

Such a parity of even-length 1-bit runs and hence the GRS sequence arises as the "dX,dY" change for each segment of the alternate paper folding curve. See "dX,dY" in Math::PlanePath::AlternatePaper.

Values Type

Parameter values_type => '0,1' gives values 0 and 1, being the count of adjacent 11s taken modulo 2, so 0 if even, 1 if odd.

values_type => '0,1'
0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, ...

FUNCTIONS

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

$seq = Math::NumSeq::GolayRudinShapiro->new ()

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

Create and return a new sequence object. The values_type parameter (a string) can be

"1,-1"        1=even, -1=odd
"0,1"         0=even, 1=odd

Random Access

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

Return the $i'th value from the sequence, being +1 or -1 (or per values_type) according to the number of adjacent 11 bit pairs in $i.

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

Return true if $value occurs in the sequence, which simply means $value == 1 or $value == -1. Or if values_type=>'0,1' then 0 or 1.

SEE ALSO

Math::NumSeq, Math::NumSeq::GolayRudinShapiroCumulative, Math::NumSeq::BaumSweet, Math::NumSeq::Fibbinary

Math::PlanePath::AlternatePaper

HOME PAGE

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

LICENSE

Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2017, 2019, 2020 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/>.