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Dan Eilers (https://oeis.org/wiki/User:_Dan_ Eilers), _, Oct 06 2006
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Dan Eilers (dan(AT)irvinehttps://oeis.comorg/wiki/User:Dan
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"It would be possible to find the counts for n=9 and n=10 using the method of my paper in JCD: http://users.cecs.anu.edu.au/~bdm/papers/ls_final.pdf [see link below]. For n=10 it is probably a 24-digit number. I'll explain the method I used. See the paper above for terminology.
B. D. McKay, A. Meynert, W. Myrvold, (2007), <a href="http://users.cecs.anu.edu.au/~bdm/papers/ls_final.pdf">Small latin squares, quasigroups, and loops</a>, J. Combin. Designs, 15 (2007), 98-119. <a href="https://doi.org/10.1002/jcd.20105">doi:10.1002/jcd.20105</a>
<a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>
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editing
"It would be possible to find the counts for n=9 and n=10 using the method of my paper in JCD: http://csusers.cecs.anu.edu.au/~bdm/papers/ls_final.pdf. For n=10 it is probably a 24-digit number. I'll explain the method I used. See the paper above for terminology.