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

1,7

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.

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

Table of n, a(n) for n=1..22.

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))

Example

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.

Crossrefs

Cf. A129947.

Sequence in context: A165339 A131601 A309541 * A044138 A043394 A044519

Adjacent sequences: A174383 A174384 A174385 * A174387 A174388 A174389

Keyword

hard,more,nonn

Author

Geoffrey H. Morley, Mar 18 2010

Extensions

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

Status

approved