[seqfan] Splitting the square into equal parts

Jean-Luc Manguin jean-luc.manguin at unicaen.fr
Mon Nov 9 14:17:04 CET 2020

Hello everyone, 

I would like to know if this problem has been studied, because I don't find any corresponding sequence in OEIS. 
Here it is : suppose we want to split a square into equal parts ( = equal area ), with horizontal or vertical line segments, drawing the segments one by one. Every new segment must lean on two existing segments, or in other words its begin and end points must be on existing segments (of course, the first segment leans on 2 opposite edges of the square). 
Thinking about this problem as for polyominoes, with "fixed", "one-sided" and "free" items, implies to take account of symmetries. 
For one-sided and free items, the results are 
a(2) = 1 (obvious) 
a(3) = 2 
a(4) = 7 (free : 6) 
a(5) = 23 (free : 17) 
a(6) = 95 (free : 62) 
a(n+1)/a(n) seems to have a limit which is above 5.2 . 

Does anyone know something about this problem or another similar question ? 
Thank you for your attention, and for your comments. 


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