Sequence of sequences

[Richard Guy]

Dear all,
          Alex Fink, Mark Krusemeyer & I are still
trying to get our paper ``Partitions with Parts
Occurring at Most Thrice'' published.  When we've
cleaned up the bit arising from the present
message, I'll be happy to send the current preprint
version to anyone interested.

          The columns of our Table 9 (you can
reconstruct this from the info below) form a 2-way
infinite sequence of sequences, for which I use
the parameter  r:

For  r = -16, -15, -14, ..., 14, 15,  they are

?  ?  ?  A001570  A085260  A077417  A078922  A072256
A070998  A070997  A049685  A001653  A004253  A001835
A001519  A000012  A011655     ?     A057079  A005408
A002878  A001834  A030221  A002315  A033890  A057080
A057081  A054320  A097783  A077416     ?     A028230

They're probably in an array somewhere, but I have
some comments/queries.

For  r  positive, I want to have alternating signs,
so you may want to change the sign of  x  or of  r
sometimes [ and perhaps forget  (-)^n ]

The generating function is  (1-x)/(1-(r+1)x+x^2)

[this is missing in one or two cases, and given
in a slightly-more-complicated-than-necessary form
in others]

The recurrence relation is  a(n)=(1-r)a(n-1)-a(n-2)
[modulo the remark about the sign of  r  above]

The terms are solutions of the Bramahgupta equation

           (r-3)x^2 - (r+1)y^2 = -4

in which you can swap  r  with  2-r  and in which
the  x  come from the  r  sequence and the  y  from
the  2-r  sequence.  This works for ALL  r, though
the result is not too earth-shaking for  -2 < r < 4

They are values of Chebyshev polynomials.  I don't
know what a Chebyshev polynomial is, and I'm at
home, away from works of ref, but I believe that
the present values are

          (-)^n S(2n,v) = (-)^n U(2n,v/2)

where  v = sqrt(r+1).  Again, I believe that this
holds for all  r.  Also, swapping  r  with  2-r,
I believe that for  r  odd and negative (and
possibly for other  r ?)

                        = (-)^n T(2n+1,w) / w

where  w = sqrt(3-r)/2  and I'm not quite sure
if that              2  is under the sqrt sign
or not !   Comments eagerly awaited by    R.