# [seqfan] Re: A045655

franktaw at netscape.net franktaw at netscape.net
Sun Jan 1 21:41:44 CET 2012

```As I understand the definition of A045655, you take a string of length
2*n, say (for n=3) 001101. Complement it bit-wise: 110010, then reverse
it left-to-right: 010011. Now, is this rotationally equivalent to the
original string? In this case, yes: rotate left two places and you get
back 001101.

The comment by Geoffrey Critzer may be correct, but it is not at all
obvious. In my opinion, if it is correct, at least a sketch of the
proof ought to be included or referenced.

-----Original Message-----
From: Maximilian Hasler <maximilian.hasler at gmail.com>

it does not matter whether you add "reversed complement" or not,
the count will be the same whatever bijection (from { 0,...,2^n-1 }
into itself) you apply to the first and/or second component of the
pairs.

as far as I understand, the "rotations" are to be taken digit-wise in
n-bit binary words,
e.g. 011 > 110 > 101.

Maximilian

On Sun, Jan 1, 2012 at 1:18 AM, Ed Jeffery <lejeffery7 at gmail.com> wrote:
> David,
>
> It seems as if Geoffrey Critzer contradicted your definition in your
title
> for A045655 by dropping your reference to "reversed complement." The
> question is, what do you mean by that terminology: are you referring
to
> "ones complement" from binary arithmetic? If so, then for longer
strings
> the symmetry you seem to be suggesting will be lost.
>
> If you are referring to dihedral symmetry, then, as you know, two
objects
> in the plane are either (or they are not) congruent up to rotations
or they
> are (or are not) reflections in a line. So, in terms of sequences of
digits:
>
> Are you ordered pairs (a,b) supposed to be such that either (i) a
equals b,
> or (ii) b is a reflection of a (i.e., b takes the digits of a in
reverse
> order)? If so, then evidently Geoffrey Critzer's definition must be
the
> correct one.
>
> It is a bit confusing, but I like your sequence.
>
> Regards,
>
> Ed Jeffery
>
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>
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