A217374 -id:A217374 - cp's OEIS frontend

A110148 Number of perfect squared rectangles of order n up to symmetries of the rectangle and of its subrectangles if any.

0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 38, 127, 408, 1375, 4783, 16645, 58059, 203808, 722575

Offset: 1

Tanya Khovanova, Feb 18 2007

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares. [Geoffrey H. Morley, Oct 12 2012]

C. J. Bouwkamp, On the dissection of rectangles into squares (Papers I-III), Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, Paper I, 49 (1946), 1176-1188 (=Indagationes Math., v. 8 (1946), 724-736); Paper II, 50 (1947), 58-71 (=Indagationes Math., v. 9 (1947), 43-56); Paper III, 50 (1947), 72-78 (=Indagationes Math., v. 9 (1947), 57-63). [Paper I has terms up to a(12) and an incorrect value for a(13) on p. 1178.]
C. J. Bouwkamp, On the construction of simple perfect squared squares, Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, 50 (1947), 1296-1299 (=Indagationes Math., v. 9 (1947), 622-625). [Correct terms up to a(13) on p. 1299.]
I. M. Yaglom, How to dissect a square? (in Russian), Nauka, Moscow, 1968. In djvu format (1.7M), also as this pdf (9.5M). [Terms up to a(13) on pp. 26-7.]
Index entries for squared rectangles
Index entries for squared squares

Cf. A217154 (counts symmetries of any subrectangles as distinct).

Cf. A181735, A217153, A217156.

a(n) = A002839(n) + A217152(n) + A217374(n). - Geoffrey H. Morley, Oct 12 2012

a(n) = a(n-1) + A002839(n) + A002839(n-1) + A217152(n) + A217152(n-1). - Geoffrey H. Morley, Oct 12 2012

Definition corrected and a(14)-a(19) added by Geoffrey H. Morley, Oct 12 2012

A217375 Number of trivially compound perfect squared rectangles of order n up to symmetries of the rectangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 40, 168, 604, 2076, 7320, 26132, 93352, 333992, 1199716, 4329180

Offset: 1

Geoffrey H. Morley, Oct 02 2012

A squared rectangle is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares.

A squared rectangle is simple if it does not contain a smaller squared rectangle, compound if it does, and trivially compound if a constituent square has the same side length as a side of the squared rectangle under consideration.

I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 24. [A217153 up to a(18).]
Index entries for squared rectangles

Cf. A217374 (counts symmetries of squared subrectangles as equivalent).

Cf. A217154.

a(n) >= 2*a(n-1) + 4*A002839(n-1) + 4*A217153(n-1), with equality for n<19.

a(20) corrected by Geoffrey H. Morley, Oct 12 2012

cp's OEIS Frontend

A110148 Number of perfect squared rectangles of order n up to symmetries of the rectangle and of its subrectangles if any.

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