OFFSET
1,8
COMMENTS
A squaring of order n is a dissection of a square into exactly n smaller squares.
This sequence counts geometric squarings up to the dihedral symmetries of the outer square.
A proper square patch is a union of whole tiles whose union is a square strictly smaller than the outer square.
A squaring is square-patch-reducible if it contains a proper square patch that, after translation, rescaling, and possibly applying a symmetry of the square, is itself a nontrivial squaring.
Otherwise it is square-patch-irreducible.
This is weaker than the classical notion of a simple squared square or squared rectangle, which forbids proper rectangular subpatches as well as proper square subpatches.
For n <= 11 this classification is exhaustive, because all squarings up to symmetry are known exactly in that range, and every proper square patch in an order-n squaring has smaller order < n.
For n <= 11, a(n) equals the number of squarings of order n, up to symmetry, that are square-patch-irreducible; that is, they contain no proper square patch congruent, after rescaling, to a nontrivial smaller squaring.
Empirically, for n = 1..11, a(n) = A221841(n) - r(n), where r(n) is the number of square-patch-reducible squarings of order n up to symmetry.
LINKS
Stuart Anderson, Simple Imperfect Squared Rectangles.
R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, The dissection of rectangles into squares, Duke Math. J. 7 (1940), 312-340; alternative link.
EXAMPLE
a(4)=1 because the unique order-4 squaring has no proper square patch that is itself, after rescaling, a nontrivial smaller squaring.
a(8)=6 because all 6 squarings of order 8 are square-patch-irreducible.
a(9)=9 because 9 of the 16 squarings of order 9 are square-patch-irreducible.
a(11)=114 because 114 of the 183 squarings of order 11 are square-patch-irreducible.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Vicente Valle, Apr 03 2026
STATUS
approved