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%I #38 Jan 05 2018 03:00:17
%S 1,1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,2,2,3,1,1,1,2,2,3,4,2,2,2,3,3,3,
%T 4,4,1,3,3,3,4,4,5,5,6,1,1,4,4,4,5,6,6,7,8,2,2,3,5,5,7,7,8,8,9,11,1,3,
%U 3,4,7,7,7,10,11,12,12,13
%N Sizes of triangles in the tilings of a convex polygon by n equilateral triangles described in A014529 written as triangle T(n,m), n >= 1, 1 <= m <= n.
%C Sizes are counted as multiples of the side length of a smallest triangle with assumed side length 1.
%C If more than one tiling resulting in a polygon of maximum area exists for a given number of triangles, the lexicographically smallest list of triangles is given.
%C The sequence is based on an illustration given in the German translation of Ian Stewart's article in Math. Recreations, Scientific American, Jul 15 1997, p. 96.
%C A continuation of the sequence would need the details of _John W. Layman_'s extension of A014529.
%C There is a tiling with 12 equilateral triangles and _John W. Layman_'s calculated area of 860 that has triangles of sizes 1, 3, 3, 4, 7, 7, 7, 10, 11, 12, 12, 13. - _Peter Munn_, Aug 23 2017
%C There is a tiling with 13 equilateral triangles and _John W. Layman_'s calculated area of 1559 that has 13 triangles of sizes 2, 2, 3, 5, 7, 7, 7, 12, 14, 15, 15, 16, 18. - _Peter Munn_, Aug 24 2017
%C From _Peter Munn_, Jan 01 2018: (Start)
%C There are tilings matching the other areas calculated by _John W. Layman_ as follows:
%C A hexagon of area 2831 tiled with 14 equilateral triangles of sizes 1, 4, 5, 5, 6, 6, 11, 11, 16, 17, 20, 20, 23, 24 with the smallest 5 arranged as in the A014529 tiling with 9 triangles, and the size 24 triangle placed at the concave vertex of the region tiled by the smallest 5.
%C A hexagon of area 5114 tiled with 15 equilateral triangles of sizes 1, 1, 6, 6, 7, 8, 8, 15, 15, 21, 23, 27, 27, 31, 32 with the smallest 6 arranged as in the A014529 tiling with 10 triangles, and the size 32 triangle placed at the concave vertex of the region tiled by the smallest 6.
%C (End)
%H Ian Stewart, <a href="http://www.spektrum.de/magazin/die-unscheinbare-schwester-der-goldenen-zahl/824251">Die unscheinbare Schwester der goldenen Zahl</a>, Spektrum der Wissenschaft, Dossier 02/2003: Mathematische Unterhaltungen II, 55-57.
%e Triangle starts:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 1, 1, 2;
%e 1, 1, 1, 2, 2;
%e 1, 1, 1, 2, 2, 3;
%e 1, 1, 1, 2, 2, 3, 4;
%e 2, 2, 2, 3, 3, 3, 4, 4;
%e 1, 3, 3, 3, 4, 4, 5, 5, 6;
%e 1, 1, 4, 4, 4, 5, 6, 6, 7, 8;
%e ...
%e For corresponding illustrations see A014529.
%Y Cf. A014529.
%K nonn,tabl,more
%O 1,10
%A _Hugo Pfoertner_, Jul 16 2017
%E Definition and data changed to resolve problem described in discussion dated Jul 19 2017 by _Hugo Pfoertner_, Aug 23 2017
%E Terms corresponding to A014529(12) from _Peter Munn_, Aug 23 2017
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