A217154 Number of perfect squared rectangles of order n up to symmetries of the rectangle.

0, 0, 0, 0, 0, 0, 0, 0, 2, 14, 62, 235, 821, 2868, 10193, 36404, 130174, 466913, 1681999, 6083873

Offset: 1

Author

Geoffrey H. Morley, Sep 27 2012

Keywords

• nonn
• hard
• more

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

• Index entries for squared rectangles
• Index entries for squared squares

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