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%I #34 Oct 05 2020 05:57:25
%S 1,1,1,4,1,2,1,8,4,21,29
%N a(n) is the number of essentially different optimal solutions with A336660(n) 1 X 1 cells to R. Kurchan's n-squares puzzle 578.
%C Numbers a(n) are a by-product of the exhaustive computation done for A336660. They were communicated privately by _Hugo Pfoertner_ and Hermann Jurksch.
%C For historical background see link section "Pictures of all different solutions for n <= 11".
%H Rainer Rosenthal, <a href="/A337515/a337515_1.gif">Illustration of a(4)</a>.
%H Rainer Rosenthal, <a href="/A337515/a337515.gif">Illustration of a(8)</a>.
%H Rainer Rosenthal, <a href="/A337515/a337515.txt">Data of all different solutions for n <= 11</a>, October 2020.
%H Rainer Rosenthal, <a href="/A337515/a337515.pdf">Pictures of all different solutions for n <= 11</a>, October 2020.
%e The square with side length s has lower left corner (x_s,y_s).
%e The a(8) = 8 solutions come in 6 main variations:
%e x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1
%e V1 0 0 0 2 4 6 5 5 2 7 6 4 1 7 4 10
%e V2 0 0 2 3 7 0 5 2 7 5 6 1 8 8 10 7
%e V3+ 0 0 3 7 0 3 2 6 5 5 1 7 7 4 3 6
%e + .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 8
%e V4+ 0 0 3 7 0 3 2 6 5 6 1 7 6 9 4 10
%e + .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 8
%e V5 0 0 4 6 1 3 5 5 2 7 6 4 3 10 2 9
%e V6 0 0 4 6 5 5 2 4 6 3 3 7 7 2 9 4
%e The two solutions in V3 are "1-siblings", i.e., they are equal except for (x1,y1).
%e Likewise V4 consists of two 1-siblings. See illustration of a(8) in the link section.
%Y Cf. A336659, A336660, A336782.
%K nonn,hard,more
%O 1,4
%A _Rainer Rosenthal_, Sep 19 2020
%E a(4) and a(8) corrected by _Rainer Rosenthal_, Oct 02 2020
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