cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A217154 Number of 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, 2, 14, 62, 235, 821, 2868, 10193, 36404, 130174, 466913, 1681999, 6083873
Offset: 1

Views

Author

Geoffrey H. Morley, Sep 27 2012

Keywords

Comments

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.
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.

Examples

			a(10) = 14 comprises the A002839(10) = 6 simple perfect squared rectangles (SPSRs) of order 10 and the 8 trivially compound perfect squared rectangles which each comprises one of the two order 9 SPSRs and one other square.

References

  • See crossrefs for references and links.

Links

Crossrefs

Cf. A110148 (counts symmetries of any squared subrectangles as equivalent).
Cf. A181735, A217156.

Formula

a(n) = A002839(n) + A217153(n) + A217375(n).
a(n) >= 2*a(n-1) + A002839(n) + 2*A002839(n-1) + A217153(n) + 2*A217153(n-1), with equality for n<19.

Extensions

a(19) and a(20) corrected by Geoffrey H. Morley, Oct 12 2012

A217374 Number of trivially compound perfect squared rectangles of order n up to symmetries of the rectangle and its subrectangles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 16, 60, 194, 622, 2128, 7438, 25852, 90266, 317350, 1127800
Offset: 1

Views

Author

Geoffrey H. Morley, Oct 02 2012

Keywords

Comments

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.

Links

  • 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(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).
  • Index entries for squared rectangles

Crossrefs

Cf. A217375 (counts symmetries of squared subrectangles as distinct).
Cf. A110148.

Formula

a(n) = a(n-1) + 2*A002839(n-1) + 2*A217152(n-1).

Extensions

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