NAME

Math::NumSeq::MephistoWaltz -- Mephisto waltz sequence

SYNOPSIS

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

DESCRIPTION

This sequence is the Mephisto waltz, being the count mod 2 of ternary digit 2s in i.

0,0,1, 0,0,1, 1,1,0, ...            (A064990)
starting i=0

i=0 has no 2s so value=0, and likewise i=1 value=0. Then i=2 has one 2 so value=1.

The sequence can also be expressed as a morphism starting from single term 0 and repeatedly expanding

0 -> 0,0,1
1 -> 1,1,0

So

0
0,0,1
0,0,1, 0,0,1, 1,1,0,
0,0,1, 0,0,1, 1,1,0, 0,0,1, 0,0,1, 1,1,0, 1,1,0, 1,1,0, 0,0,1

|   original      |  |     copy        |  |  0<->1 flipped  |
+-----------------+  +-----------------+  +-----------------+

The effect of the expansion is keep the first third the same, append a copy of it, and append an 0<->1 flipped copy of it .

FUNCTIONS

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

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

Create and return a new sequence object.

Random Access

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

Return the $i'th MephistoWaltz value, being the count mod 2 of the ternary digit 2s in $i.

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

Return true if $value occurs in the sequence, which simply means 0 or 1.

FORMULAS

The calculation can also be made in a power-of-3 base like 9, 27, 81, etc instead of just 3. For example in base 9 digits 2, 5, 6, 7 have a one (mod 2) ternary 2. These base 9 digits correspond to the 1s in the initial sequence 0,0,1, 0,0,1, 1,1,0 shown above.

SEE ALSO

Math::NumSeq, Math::NumSeq::DigitSumModulo

HOME PAGE

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

LICENSE

Copyright 2011, 2012, 2013, 2014, 2016, 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/>.