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Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 1 element.
5

%I #30 Dec 15 2022 10:00:30

%S 1,2,2,4,4,12,8,44,32,228,148,1632,912,16004,8420,213680,101508,

%T 3933380,1691008,98949060,38742844,3413919788,1213540776,161410887252,

%U 52106993880

%N Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 1 element.

%C From _Walter Trump_, Dec 15 2022: (Start)

%C a(n) is the number of fully symmetric dissections of an n X n square into squares with integer sides.

%C Conjecture: For n>3 the number of dissections is a multiple of 4. (End)

%H Christopher Hunt Gribble, <a href="/A226978/a226978.txt">C++ program for A226978, A226979, A226980, A226981, A227004</a>

%H Walter Trump, <a href="/A226978/a226978.png">Example for n=19</a>

%F a(n) + A226979(n) + A226980(n) + A226981(n) = A224239(n).

%F 1*a(n) + 2*A226979(n) + 4*A226980(n) + 8*A226981(n) = A045846(n).

%e For n=5, there are 4 dissections where the orbits under the symmetry group of the square, D4, have 1 element.

%e For n=4, 3 dissections divide the square into uniform subsquares (of sizes 1, 2 and 4 respectively), and this is the 4th:

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%Y Cf. A045846, A034295, A219924, A224239, A226979, A226980, A226981.

%K nonn,more

%O 1,2

%A _Christopher Hunt Gribble_, Jun 25 2013

%E a(8)-a(12) from _Ed Wynn_, Apr 02 2014

%E a(13)-a(25) from _Walter Trump_, Dec 15 2022