[seqfan] Re: A045655

[Olivier Gerard]

Hello,

The comment by Geoffrey is indeed correct but his various programs were
inefficient or relying on obsolete librairies not loaded (i.e.
Combinatorica).

I have replaced it by another version and added a few comments.

There should be a formula in the form of a sum of squares giving the
general term of the sequence instead of brute force enumeration.

Olivier


On Sun, Jan 1, 2012 at 21:41, <franktaw at netscape.net> wrote:

> 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.
>
> Franklin T. Adams-Watters
>
>
> -----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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