A174386 Smallest possible area for a perfect isosceles right triangled square of order n; or 0 if no such square exists.

0, 0, 0, 0, 0, 0, 49, 98, 121, 196, 128, 196, 289, 242, 441, 441, 484, 722, 722, 1024, 1156, 1225

Offset: 1

Author

Geoffrey H. Morley, Mar 18 2010

Keywords

• hard
• more
• nonn

Comments

These terms are only conjectures, but are thought highly likely to be correct for n<18.

A tiling is perfect if no two tiles are the same size. The order of a tiling is the number of tiles. An integer n is the order of a perfect isosceles right triangled square if and only if n>=7.

We require the area of each triangular tile to be m^2 or m^2/2, where m is an integer. A tiling of an integer-sided square can often be scaled by the factor 1/sqrt(2) and still meet this requirement.

Examples

			Diagrams for the known tilings associated with the conjectured terms up to a(20) are in pdfs downloadable from Stuart Anderson's website. All squares are depicted with integer sides, but many can be scaled by the factor 1/sqrt(2).
In MIRT code (see link for an explanation) one of the three known tilings for n=21, area 1156, is -162 114 66 127 126 -56 82 -113 102 -63 -83 35 34 -103 -57 90 91 -26 45 44 -25.
The known tiling for n=22, area 1225, is -165 134 103 92 91 -76 -36 67 143 47 46 -52 -85 84 -23 75 -14 53 -120 121 -115 114.
The tilings for a(11) and a(12) were found by J. D. Skinner and published in Skinner et al. (2000). G. H. Morley found the tilings for subsequent terms.
		

References

• J. D. Skinner II, C. A. B. Smith, and W. T. Tutte, On the Dissection of Rectangles into Right-Angled Isosceles Triangles, Journal of Combinatorial Theory, Series B 80 (2000), 277-319.

Links

• S. E. Anderson, Perfect Squared Rectangles, Squared Squares, and Isosceles Right Triangled Squares
• S. E. Anderson, Tilings by Triangles (see Morley's Isosceles Right Triangulation code (MIRT code))

Crossrefs

Cf. A129947.

Extensions

XB code renamed MIRT code by Geoffrey H. Morley, May 12 2012