A005097 (Odd primes - 1)/2 implicitly referenced in new arXiv paper

N. J. A. Sloane njas at research.att.com
Wed Jun 11 02:52:09 CEST 2008

On Sun, 18 May 2008, Richard Guy wrote about a families
of third-order & fourth-order (``symmetric'') sequences.
I believe that someone took this up and ran with it.

Here is another opportunity, which also raises some old
questions.  Let me start from  A015455, about which more
could be written.

First of all, there's the old problem of offset, which I
shall never succeed in learning about.  Should the sequence
not start  0, 1, 1, 10, ... ?

There should be a generating function, which is
where the something might be  1+x  or  1-8x  or  1-8x-2x^2
depending on my algebra, and which part of the sequence
we're talking about.

Because the second old problem is that of the doubly
infinite sequence.  In the context in which I came across
it, it was
... 502741, 55187, 6058, 665, 73, 8, 1, 1, 10, 91, 829, ...
(no zero!) and  1, 8, 73, 665, 6058, ... is not in OEIS
(modulo my mother's description of me as not a good looker).

Perhaps I'd better come clean and admit that the sequence
is really
..., -502741, 55187i, 6058, -665i, -73, 8i, 1, i, -10,
    -91i, 829, 7552i, -68797, -626725i, ...
and illustrates another problem, that of `signed' sequences,
except that here the signs are fourth roots of one
rather than square roots.

Where did I get it from?  By reading the labels in the
regions contiguous with the  3i  region in a diagram
of Gaussian Markov numbers.  And there are lots more
regions to read round!  And lots more diagrams because,
if we put  x = 0  in the Markov equation
               x^2 + y^2 + z^2 = 3xyz
we get  y^2 + z^2 = 0,  which doesn't have any real
solutions, other than (0,0), but has plenty of
Gaussian integer solutions, yielding Markov triples
(0,1,i), (0,2,2i), (0,3,3i), ... (0,1+i,1-i), ...
with sequences

..., 96050, -2666i, -74, 2i, 2, 70i, -2522, -90862i, ...

      ..., -19929i, -246, 31, 3, 240i, -19443, ...

      ..., -83524i, -580, 4i, 4, 572i, -81220, ...

    ..., -254255i, -1130, 5i, 5, 1120i, -251205, ...

where one should, perhaps, take the absolute values,
and divide through by 2, 3, 4, 5, ..., but I didn't
find them in OEIS, although I believe that they
are second-order recurring sequences.

..., 5473-6119i, 305-341i, 17-19i, 1-i, 1+i, 17+19i, 305+341i, ...

Here the real & imaginary parts should be in -- aha!!
bingo!! they're   A007805 & A049629  though the present
context isn't mentioned, of course!

Lots of work for someone!!      R.

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