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

1,7

Comments

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

Links

Table of n, a(n) for n=1..20.

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.

Example

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

Crossrefs

Cf. A167123, A290697, A290821.

Sequence in context: A298407 A227252 A274495 * A318231 A242271 A056198

Adjacent sequences: A299702 A299703 A299704 * A299706 A299707 A299708

Keyword

nonn,hard,more

Author

Peter Munn and Hugo Pfoertner, Feb 17 2018

Extensions

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

Status

approved