minimal perimeter of a polyhex

Sascha Kurz sascha.kurz at uni-bayreuth.de
Wed Mar 5 08:37:41 CET 2008

Dear SeqFans, dear Tanya,

the exact formula for this sequence is 2\lceil\sqrt{12 n-3}\rceil

It can be easily derrived from the minimum number of edges of a polyhex
consisting on n cells. This number was obtained by H. Harborth and is
given by 3n+\lceil\sqrt{12 n-3}\rceil.

The set of extremal examples were described by Y. S. Kupitz in "On the
maximal number of appearances of the minimal distance among $n$ points in
the plane." [Intuitive geometry. Proceedings of the 3rd international
conference held in Szeged, Hungary, from 2 to 7 September, 1991.
Amsterdam: North-Holland. Colloq. Math. Soc. János Bolyai. 63, 217-244]

It should be possible to count the maximal examples using generating
functions as it was done for squares.

Best regards,
Sascha

Am Di, 4.03.2008, 15:20, schrieb Tanya Khovanova:
> Dear SeqFans,
>
>
> I just submitted the sequence:
>
>
> 6, 10, 12, 14, 16, 18, 18, 20, 22, 22, 24, 24,
> Minimal perimeter of a polyhex with n cells.
>
>
> I am surprised that this sequence wasn't there.
>
>
> I calculated it manually. Anyone has a program dealing with polyhexes?
> Can someone extend it? Is there are program for polyhexes somewhere
>
> Best, Tanya
>
>

Dear SeqFans,

I implemented following generalization of the Thue Morse sequence :
Let Sk(n) be sum of digits of n; n in base-k notation.
Let F(t) be some arithmetic function.
Then the sequence a(n) = F(Sk(n)) mod m is a generalised Thue-Morse sequence.
The Thue-Morse sequence (A010060)  has k=2, m=2, F(S2(n)) = S2(n) , so a(n)= S2(n) mod 2.

At the moment I am working on [c/d]-Thue-Morse sequences of the type
a(n) = floor( c*S2(n) / d) mod 2 ; c,d integers;  gcd(c,d)=1;  c from < 1; 2*d-1 > ;  (S2(n) sum of digits of n in binary notation.
Are at least some of [c/d]-Thue-Morse sequences overlap free ? square free?
Sequences of partial sums of [c/d]-Thue-Morse sequences seems to be self-similar, are they fractal ?

Interpreting a(n) = floor( c*S2(n) / d) mod 2 as a binary number of the form
0 , a(0)a(1)a(2)....   and written in decimal gives [c/d]-Thue-Morse constants.
[1/1]-Thue Morse constant 0,824908067.......    is a transcendental number
(http://mathworld.wolfram.com/Thue-MorseConstant.html)

Are [c/d]-Thue-Morse constants transcendental numbers too ?

So far as I know, nothing is known about  [c/d]-Thue-Morse sequences,
but maybe somebody did seen something which could help in literature,articles etc.

Ctibor O. Zizka
Czech Republic