Kevin Ryde > Math-NumSeq > Math::NumSeq::NumAronson

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NAME ^

Math::NumSeq::NumAronson -- numerical version of Aronson's sequence

SYNOPSIS ^

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

DESCRIPTION ^

This is a self-referential sequence by Cloitre, Sloane and Vandermast,

"Numerical Analogues of Aronson's Sequence", http://arxiv.org/abs/math.NT/0305308

The sequence begins

    1, 4, 6, 7, 8, 9, 11, 13, ...
    starting i=1

Starting with a(1)=1 the rule is "n is in the sequence iff a(n) is odd". The result is a uniform pattern 3 steps by 1 then 3 steps by 2, followed by 6 steps by 1 and 6 steps by 2, then 12, 24, 48, etc.

    1,
    4,
    6,  7,  8,                   # 3 steps by 1
    9,  11, 13,                  # 3 steps by 2
    15, 16, 17, 18, 19, 20,      # 6 steps by 1
    21, 23, 25, 27, 29, 31,      # 6 steps by 2
                                 # 3*2^k steps by 1
                                 # 3*2^k steps by 2

In general

    numaronson(9*2^k-3+j) = 12*2^k - 3 + (3*j+abs(j))/2
    where -3*2^k <= j < 3*2^k

The (3*j+abs(j))/2 part is the step, going by 1 if j<=0 and by 2 if j>0.

FUNCTIONS ^

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

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

Create and return a new sequence object.

Random Access

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

Return the $ith value in the sequence.

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

Return true if $value occurs in the sequence.

SEE ALSO ^

Math::NumSeq, Math::NumSeq::Aronson

HOME PAGE ^

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

LICENSE ^

Copyright 2010, 2011, 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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