A289944 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.

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, 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, 3, 4, 7, 7, 7, 10, 11, 12, 12, 13

Offset: 1

Author

Hugo Pfoertner, Jul 16 2017

Keywords

• nonn
• tabl
• more

Comments

Sizes are counted as multiples of the side length of a smallest triangle with assumed side length 1.

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.

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.

A continuation of the sequence would need the details of John W. Layman's extension of A014529.

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

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

From Peter Munn, Jan 01 2018: (Start)

There are tilings matching the other areas calculated by John W. Layman as follows:

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.

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.

(End)

Examples

			Triangle starts:
  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, 4, 4;
  1, 3, 3, 3, 4, 4, 5, 5, 6;
  1, 1, 4, 4, 4, 5, 6, 6, 7, 8;
...
For corresponding illustrations see A014529.
		

Links

• Ian Stewart, Die unscheinbare Schwester der goldenen Zahl, Spektrum der Wissenschaft, Dossier 02/2003: Mathematische Unterhaltungen II, 55-57.

Crossrefs

Cf. A014529.

Extensions

Definition and data changed to resolve problem described in discussion dated Jul 19 2017 by Hugo Pfoertner, Aug 23 2017

Terms corresponding to A014529(12) from Peter Munn, Aug 23 2017