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

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Module Version: 72

# 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.

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