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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. 3
1, 0, 0, 2, 0, 3, 4, 4, 6, 5, 8, 6, 6, 7, 8, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

LINKS

Table of n, a(n) for n=1..16.

Stuart Anderson, An Introduction to Triangled Equilateral Triangles

Ales Drapal, Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.

Hugo Pfoertner, Illustration for a(16)=7.

EXAMPLE

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:

                *

               / \

              +   +

             /     \

            *---+---*

           / \     / \

          +   +   *---*

         /     \ / \ / \

        *---*---*---*---*

       / \ / \         / \

      *---*   +       +   +

     / \ /     \     /     \

    *---*       +   +       +

   / \ /         \ /         \

  *---*---+---+---*---+---+---*

CROSSREFS

Cf. A167123, A290653, A290697, A290821.

Sequence in context: A137218 A306765 A087819 * A278029 A066246 A198370

Adjacent sequences:  A290817 A290818 A290819 * A290821 A290822 A290823

KEYWORD

nonn,more

AUTHOR

Hugo Pfoertner, Aug 11 2017

EXTENSIONS

Title changed as suggested by Peter Munn, Feb 17 2018

STATUS

approved

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Last modified March 30 19:49 EDT 2020. Contains 333127 sequences. (Running on oeis4.)