A299705 Number of ways to dissect an equilateral triangle into n non-overlapping equilateral triangles counting isomorphisms only once.

1, 0, 0, 1, 0, 1, 2, 3, 9, 23, 62, 188, 574, 1826, 5953, 19664, 66049, 224700, 771859, 2674753

Offset: 1

Author

Peter Munn and Hugo Pfoertner, Feb 17 2018

Keywords

• nonn
• hard
• more

Comments

Data from Figure 7 of Drapal and Hamalainen, see link.

Examples

			a(9)=9:
        *                 *                              *
       / \               / \                            / \
      *---*             +   +                          +   +
     / \ / \           /     \                        /     \
    *---*---*         +       +                      +       +
   / \ / \ / \       /         \                    /         \
  *---*---*---*     +           +                  +           +
                   /             \                /             \
                  *---+---*---+---*              *---+---+---*---*
                 / \     / \     / \            / \         / \ / \
                +   +   +   +   +   +          +   +       *---*---*
               /     \ /     \ /     \        /     \     / \     / \
              +       *---*---*       +      +       +   +   +   +   +
             /         \ / \ /         \    /         \ /     \ /     \
            *---+---+---*---*---+---+---*  *---+---+---*---+---*---+---*
.
              *         *---+---+---*---+---+---*         *
             / \         \         / \         /         / \
            +   +         +       +   +       +         +   +
           /     \         \     /     \     /         /     \
          +       +         +   *---*---*   +         +       +
         /         \         \ / \ / \ / \ /         /         \
        +           +         *---*---*---*         +           +
       /             \         \         /         /             \
      *---+---*---+---*         +       +         *---+---*---*---*
     / \     / \     / \         \     /         / \     / \ / \ / \
    +   +   +   +   *---*         +   +         +   +   +   *---*   +
   /     \ /     \ / \ / \         \ /         /     \ /     \ /     \
  *---+---*---+---*---*---*         *         *---+---*---+---*---+---*
.
              *         *---+---*---*---*---+---*         *
             / \         \     / \ / \ / \     /         / \
            +   +         +   +   *---*   +   +         +   +
           /     \         \ /     \ /     \ /         /     \
          *---*---*         *---+---*---+---*         +       +
         / \ / \ / \         \             /         /         \
        *---*---*---*         +           +         *---+---+---*
       / \         / \         \         /         / \         / \
      +   +       +   +         +       +         +   +       +   +
     /     \     /     \         \     /         /     \     /     \
    +       +   +       +         +   +         *---*---*   +       +
   /         \ /         \         \ /         / \ / \ / \ /         \
  *---+---+---*---+---+---*         *         *---*---*---*---+---+---*
		

Links

• Ales Drapal, Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.
• Hugo Pfoertner, Rainer Rosenthal, Illustration of the 23 dissections into 10 triangles.

Crossrefs

Cf. A167123, A290697, A290821.

Extensions

Offset changed, also name to accommodate, following suggestion by M. F. Hasler, Feb 23 2018