[seqfan] Question re: A250000

[Bob Selcoe]

Hello Seqfans,

The sequence A250000 (maximum number of peacefully coexisting equal-sized 
"armies" of queens on chess boards of varying n X n sizes) poses some 
fascinating problems.

The length of the sequence is small; only up to a(13) = 24.

The "known" lower bound for a solution is a(n) = 9/64*n^2. There is a link 
to a paper by Prestwich and Beck referenced in the sequence which expands on 
this idea.  I can't follow the paper, but it apparently provides an upper 
bound, as well.

For all n = 4m+1 m>=0, I can show a pattern of quasi-symmetric queen 
placement such that a(n) = 2m(m+1).  For m = {0..3}, this is indeed the 
maximum number of queens possible.  For m>=4, these solutions are < the 
known lower bound of (9/64)*n^2.

I have proposed for A250000 examples of solutions using this queen pattern 
for n=9 and n=13.  Please refer to the sequence history to see the pattern. 
Since the pattern yields 40 for n=17, and a(17)=42 is the known lower bound, 
it (apparently) does not provide a solution for a(17).

Can anyone show an actual example of a 17 X 17 chessboard where the number 
of queens > 40, even if it can't be proven to be a solution (maximum number 
of queens) for a(17)?

Best Wishes,
Bob Selcoe