sum-of-(non-zero)-squares representations versus partitions into squares

[wouter meeussen]

Hi all,
here's a sketch that might yield to analysis:
consider the count of sum-of-(non-zero)-squares representation of n=64 in k terms:
k=64: 1 :: {1,1, ..,1} : the all-ones case
k=63: 0 :: nill
k=62: 0 :: nill
k=61: 1 :: {4,1,1,..,1}
k=60: 0
etc.

Now, it turns out that for n sufficiently big, the count for k downwards from n to 1 always goes:
{1,0,0,1,0,0,1,0,1,1,0,1,1,0,1,2,1,1,2,1,1,2,1,2,4,1,2,4,1,2,5,
2,4,5,2,5,5,2,6,7,4,6,7,5,6,8,6,8,12,6,9,13,6,..
where the last integers are in doubt, 'cause slowly converging to a limiting value.
It would be good to pin down the length of the 'fully converged' part. Not done yet.

It also seems evident that there must be a link to the partitions into squares.(A001156)
1,1,1,1,2,2,2,2,3,4,4,4,5,6,6,6,8,9,10,10,12,13,14,14,16,19 ..
but I don't see it straight away. Anyone care to enlighten me?

W.