

A174386


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


2



0, 0, 0, 0, 0, 0, 49, 98, 121, 196, 128, 196, 289, 242, 441, 441, 484, 722, 722, 1024, 1156, 1225
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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 integersided 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 RightAngled Isosceles Triangles, Journal of Combinatorial Theory, Series B 80 (2000), 277319.


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: A044030 A165339 A131601 * 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



