# [seqfan] Re: Project: sequences obtained from Gaussian Integers via Penney's binary method of encoding?

Joerg Arndt arndt at jjj.de
Sun Aug 21 18:54:18 CEST 2016

```I always found that the complex numeration systems with
radix 2 + i and digits 0, +1, -1, +i, and -i
should be better known (and researched).
Also, don't forget the numeration systems
on the triangular grid (see pp.21-29 of
http://arxiv.org/abs/1607.02433
for a glimpse).
Oh, and there is https://oeis.org/A265671
For the fundamental tile of the corresponding numeration
system see https://oeis.org/A265671/a265671_1.png

I have references to more than a hundred papers
marked as pertinent to the subject.
If you want them then email me off-list and
I'll fling the file into your general direction.

Best regards,   jj

* Antti Karttunen <antti.karttunen at gmail.com> [Aug 21. 2016 17:38]:
> On Sat Aug 20 23:14:57 CEST 2016 Don Reble wrote in
> http://list.seqfan.eu/pipermail/seqfan/2016-August/016647.html
>
> >> Consider the method of expressing Gaussian Integers as sums of powers of
> >> (-1 + i) (with coefficients just either 0 or 1).
>
> >    vol. 2, 2nd ed., pp 189-190.
>
> Thanks, I looked at that page (thanks to some pirates on the net).
> Knuth writes: "Another 'binary' complex number system maybe be obtained by
> using the base i-1, as suggested by W. Penney [JACM 12 (1965), 247 - 248]."
> (So maybe we should call it Penney's binary encoding, or just base (i-1)
> encoding?)
>
> The "quater-imaginary system" of Knuth (presented just before, on the same
> page 189, using base 2i) might have more regular encodings, as "... so
> conversion to and from quater-imaginary notation reduces to conversion to
> and from negative quaternary representation of the real and imaginary
> parts. The interesting property of this system is that it allows
> multiplication and division of complex number to be done in a fairly
> uniform manner without treating real and imaginary parts separately".
> (I guess Marc LeBrun's binary interleaving sequences come also handy here?)
>
>
> Best,
>
> Antti
>
>
> >--
> > Don Reble  djr at nk.ca
>
>
>
> On Sat, Aug 20, 2016 at 10:31 PM, Antti Karttunen <antti.karttunen at gmail.com>
> wrote:
> >
> >
> >
> > Here's an old idea from my attic:
> >
> > Consider the method of expressing Gaussian Integers as sums of powers of
> (-1 + i) (with coefficients just either 0 or 1).
> > I call this "Hungarian binary method", as I first saw it at this web
> page: http://szdg.lpds.sztaki.hu/szdg/desc_numsys_es.php
> > (but please tell if it is much older and discovered in some other
> country!)
> >
> > By hand calculation, I get the following values for some small binary
> vectors:
> >
> > 0000 = 0
> > 0001 = 1
> > 0010 = -1 + i
> > 0011 = i
> > 0100 = -2i
> > 0101 = 1 - 2i
> > 0110 = -1 - i
> > 0111 = -i
> > 1000 = 2 + 2i = (-1+i)^3
> > 1001 = 3 + 2i
> > 1010 = 1 + 3i
> > 1011 = 2 + 3i
> > 1100 = 2
> > 1101 = 3
> > 1110 = 1 + i
> > 1111 = 2 + i
> > 10000 = (-1+i)^4 = 1 + -4i + -6 + 4i + 1 = 2 -6 = -4
> >
> > so with these we finally get:
> > 11101 = -1 (for a moment I was afraid that it would not appear at all!)
> >
> > Would it make sense to express with this system all kind of functions
> whose domains and/or ranges are Gaussian Integers?
> > (As OEIS sequences, I mean).
> >
> > That is, in a slightly similar way as functions whose domain and/or range
> are polynomials over GF(2) have been submitted via their encoded binary
> representation, found under index entry
> http://oeis.org/wiki/Index_to_OEIS:_Section_Ge#GF2X
> >
> > Or are there better ones for Gaussian Integers? I'm not thinking here
> about how easy it is for human beings to search the terms, but in terms of
> some vaguely felt elegance/inelegance of the sequences encoded with it.
> That -1 appears so "late" (encoded by binary representation of 29) in this
> system feels a bit worrisome.
> >
> > In any case, as long as bitwise-AND(x,y) = 0 (x and y have no 1-bits in
> common positions when encoded), then Gaussian_Integer_interpretation(x+y) =
> Gaussian_Integer_interpretation(x) + Gaussian_Integer_interpretation(y).
> >
> >
> > For the starters, could somebody generate a list of numbers whose binary
> representations map via this particular encoding to Gaussian primes? (I
> don't have appropriate mathematical software at my reach, or cannot use it
> properly).
> >
> >
> >
> > Best regards,
> >
> > Antti
> >
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
```