A290820 Side length of the smallest equilateral triangles that have a separated dissection into n equilateral triangles with integer sides, or 0 if no such triangle exists.

1, 0, 0, 2, 0, 3, 4, 4, 6, 5, 8, 6, 6, 7, 8, 7

Offset: 1

Author

Hugo Pfoertner, Aug 11 2017

Keywords

• nonn
• more

Comments

No solution exists for n = [2, 3, 5].

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(6) = 3:
        *
       / \
      *---*
     / \ / \
    *---*   +
   / \ /     \
  *---*---+---*
a(7) = 4:
          *
         / \
        +   +
       /     \
      *---*---*
     / \ / \ / \
    +   *---*   +
   /     \ /     \
  *---+---*---+---*
a(8) = 4:
          *
         / \
        *---*
       / \ / \
      *---*   +
     / \ /     \
    *---*       +
   / \ /         \
  *---*---+---+---*
a(9) = 6:
              *
             / \
            +   +
           /     \
          *---+---*
         / \     / \
        +   +   +   +
       /     \ /     \
      *---+---*       +
     / \     /         \
    *---*   +           +
   / \ / \ /             \
  *---*---*---+---+---+---*
a(10) = 5:
            *
           / \
          *---*
         / \ / \
        *---*   +
       / \ /     \
      *---*       +
     / \ /         \
    *---*           +
   / \ /             \
  *---*---+---+---+---*
a(11) = 8:
                  *
                 / \
                +   +
               /     \
              *---+---*
             / \     / \
            +   +   +   +
           /     \ /     \
          *---+---*       +
         / \     /         \
        +   +   +           +
       /     \ /             \
      *---+---*               +
     / \     /                 \
    *---*   +                   +
   / \ / \ /                     \
  *---*---*---+---+---+---+---+---*
a(12) = 6:
              *
             / \
            *---*
           / \ / \
          *---*   +
         / \ /     \
        *---*       +
       / \ /         \
      *---*           +
     / \ /             \
    *---*               +
   / \ /                 \
  *---*---+---+---+---+---*
a(13) = 6:
              *
             / \
            +   +
           /     \
          *---+---*
         / \     / \
        +   +   *---*
       /     \ / \ / \
      *---*---*---*   +
     / \ / \     /     \
    *---*   +   +       +
   / \ /     \ /         \
  *---*---+---*---+---+---*
a(14) = 7:
                *
               / \
              +   +
             /     \
            +       +
           /         \
          *---+---*---*
         / \     / \ / \
        +   +   *---*   +
       /     \ / \ /     \
      *---*---*---*       +
     / \ / \     /         \
    *---*   +   +           +
   / \ /     \ /             \
  *---*---+---*---+---+---+---*
a(15) = 8:
                  *
                 / \
                +   +
               /     \
              *---+---*
             / \     / \
            +   +   *---*
           /     \ / \ / \
          +       *---*   +
         /         \ /     \
        *---+---+---*       +
       / \         /         \
      *---*       +           +
     / \ / \     /             \
    *---*   +   +               +
   / \ /     \ /                 \
  *---*---+---*---+---+---+---+---*
a(16) = 7:
                *
               / \
              +   +
             /     \
            *---+---*
           / \     / \
          +   +   *---*
         /     \ / \ / \
        *---*---*---*---*
       / \ / \         / \
      *---*   +       +   +
     / \ /     \     /     \
    *---*       +   +       +
   / \ /         \ /         \
  *---*---+---+---*---+---+---*
		

Links

• Stuart Anderson, An Introduction to Triangled Equilateral Triangles
• Ales Drapal and Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.
• Hugo Pfoertner, Illustration for a(16)=7.

Crossrefs

Cf. A167123, A290653, A290697, A290821, A300001.

Extensions

Title changed as suggested by Peter Munn, Feb 17 2018