A167123 Number of isomorphism classes of separated dissections of an equilateral triangle into n nonoverlapping equilateral triangles.

1, 0, 0, 1, 0, 1, 2, 3, 8, 20, 55, 161, 478, 1496, 4804, 15589, 51377, 172162, 583810, 1998407

Offset: 1

Author

Jonathan Vos Post, Oct 27 2009

Keywords

• nonn

Comments

A dissection into 5 triangles is impossible.

From table on p.11 of Drapal. The authors write: We enumerate all dissections of an equilateral triangle into smaller equilateral triangles. We confirm W. T. Tutte's claim that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20.

The meaning of "separated dissection" is defined at the end of the introduction of the Drapal and Hamalainen article, see link. - Hugo Pfoertner, Feb 17 2018

Examples

			a(8)=3:
            *
           / \
          /   \
         /     \
        /       \
       /         \
      *---*-------*
     / \ / \     / \
    *---*   \   /   \
   / \ /     \ /     \
  *---*-------*-------*
            *
           / \
          /   \
         /     \
        *-------*
       / \     / \
      *---*   /   \
     / \ / \ /     \
    /   *---*       \
   /     \ /         \
  *-------*-----------*
          *
         / \
        *---*
       / \ / \
      *---*   \
     / \ /     \
    *---*       \
   / \ /         \
  *---*-----------*
		

Links

• Ales Drapal, Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.
• Stuart Anderson, An Introduction to Triangled Equilateral Triangles.

Crossrefs

Cf. A290653, A290697, A299705.

Extensions

a(1)-a(3) added and name accommodated as suggested by M. F. Hasler, Feb 23 2018

Corrected and extended by Hugo Pfoertner, Aug 09 2017

Definition corrected by Hugo Pfoertner, Feb 17 2018