New twist(?) on old point-counting problem

[Jonathan Post]

I'm interested in this. I have one recent OEIS sequence that enumerates ways
to partition n into the ordered sum of squares of integers less than 5.
That was a tiny excerpt from a long monograph on "imaginary logic" in which
I find myself enumerating vectors in vector subspaces of Z^n by the
Euclidean norm | (a, b, c, d) | = a^2 + b^2 + c^2 + d^2. I find that the
norms divide in many cases the elements of my systems from "forbidden"
vectors in the same vector space, which are not elements in my *-Algebra,
closed under certain operations including string reversal and concatenation.
More generally, I'm wrestling with quaternion norms and norms in *-Algebras,
and trying not to be led astray into Banach spaces and Hopf Algebras, to the
detriment of the study's original goals, and the integer sequences which
fall out of it.

One seqfan asked for an got an earlier partial draft of my long paper, and
found that I'd failed to answer the question in which he was interested.
And that seqfan is VERY expert in the matrix theory underlying Quantum
Mechanics and entanglement.  I have sometimes offended seqfans with "ugly"
sequences, and/or too many sequences.  I have yet to earn a "nice" after
over 1,370 tries.  I depend on criticism to redirect me in the right
direction.  Criticism is always useful (not name-calling, mind you) and
editors are essential.  OEIS has an amazing editorial board, excellently
managed by njas. I have learned much from them, and much already from
seqfans.

Surely someone in seqfans knows it David Wilson has rediscovered something
known for a century, or not.

On 10/27/06, David Wilson <davidwwilson at comcast.net> wrote:
>
>  (Best viewed in fixed width)
>
> Let f be defined as
>
> f(x, y) =
>     0, if y > x
>     1, if y = 0
>     2*SUM(k >= 1 and x-k^2 >= y; f(x-k^2, y-1)), otherwise.
>
> A table of f(x, y), omitting the zero elements:
>
>    \y|    0    1    2    3    4    5    6    7    8    9   10
>    x\|
> -----+-------------------------------------------------------
>    0 |    1
>    1 |    1    2
>    2 |    1    2    4
>    3 |    1    2    4    8
>    4 |    1    4    4    8   16
>    5 |    1    4   12    8   16   32
>    6 |    1    4   12   32   16   32   64
>    7 |    1    4   12   32   80   32   64  128
>    8 |    1    4   16   32   80  192   64  128  256
>    9 |    1    6   16   56   80  192  448  128  256  512
>   10 |    1    6   24   56  176  192  448 1024  256  512 1024
>
> Now use the xth row of this table as differences to generate sequence S_x.
> For example, taking x = 3, the third row is (1 2 4 8). Using these as
> differences, we generate the sequence:
>
>                   8     8     8    ...
>                4    12    20    28    ...
>             2     6    18    38    66    ...
>    S_3 = 1     3     9    27    65    131   ...
>
> S_3 is indexed starting at 0.
>
> It appears that S_x(n) gives the number of points in Z^n with norm <=
> sqrt(x). For example, there are S_3(4) = 65 points of Z^4 norm <= sqrt(3),
> namely:
>
> (-1 -1 -1 0)
> (-1 -1 0 -1)
> (-1 -1 0 0)
> (-1 -1 0 1)
> (-1 -1 1 0)
> (-1 0 -1 -1)
> (-1 0 -1 0)
> (-1 0 -1 1)
> (-1 0 0 -1)
> (-1 0 0 0)
> (-1 0 0 1)
> (-1 0 1 -1)
> (-1 0 1 0)
> (-1 0 1 1)
> (-1 1 -1 0)
> (-1 1 0 -1)
> (-1 1 0 0)
> (-1 1 0 1)
> (-1 1 1 0)
> (0 -1 -1 -1)
> (0 -1 -1 0)
> (0 -1 -1 1)
> (0 -1 0 -1)
> (0 -1 0 0)
> (0 -1 0 1)
> (0 -1 1 -1)
> (0 -1 1 0)
> (0 -1 1 1)
> (0 0 -1 -1)
> (0 0 -1 0)
> (0 0 -1 1)
> (0 0 0 -1)
> (0 0 0 0)
> (0 0 0 1)
> (0 0 1 -1)
> (0 0 1 0)
> (0 0 1 1)
> (0 1 -1 -1)
> (0 1 -1 0)
> (0 1 -1 1)
> (0 1 0 -1)
> (0 1 0 0)
> (0 1 0 1)
> (0 1 1 -1)
> (0 1 1 0)
> (0 1 1 1)
> (1 -1 -1 0)
> (1 -1 0 -1)
> (1 -1 0 0)
> (1 -1 0 1)
> (1 -1 1 0)
> (1 0 -1 -1)
> (1 0 -1 0)
> (1 0 -1 1)
> (1 0 0 -1)
> (1 0 0 0)
> (1 0 0 1)
> (1 0 1 -1)
> (1 0 1 0)
> (1 0 1 1)
> (1 1 -1 0)
> (1 1 0 -1)
> (1 1 0 0)
> (1 1 0 1)
> (1 1 1 0)
>
>
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