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A300001 Side length of the smallest equilateral triangle that can be dissected into n equilateral triangles with integer sides, or 0 if no such triangle exists. 0
1, 0, 0, 2, 0, 3, 4, 4, 3, 4, 5, 6, 4, 5, 6, 4, 5, 6, 5, 6, 6, 5, 7, 6, 5, 7, 6, 6, 7, 6, 7, 7, 6, 7, 7, 6, 7, 7, 8, 7, 7, 8, 7, 8, 8, 7, 8, 8, 7, 8, 9, 8, 8, 9, 8, 8, 9, 8, 9, 9, 8, 9, 9, 8, 9, 9, 9, 10, 9, 9, 10, 9, 9, 10, 9, 10, 10, 9, 10, 10, 9, 10, 10, 10, 10, 10, 11, 10, 10, 11, 10, 10, 11, 10, 11, 11, 10, 11, 11, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

No solutions exist for n = 2, 3 and 5.

a(n) = A290820(n) for n <= 8. It is conjectured that a(n) < A290820(n) for all n > 12.

The seven numbers mentioned by Peter Munn in the Formula section [1, 2, 4, 5, 7, 10, 13] coincide with the seven terms of A123120. - M. F. Hasler and Omar E. Pol, Feb 23 2018

LINKS

Table of n, a(n) for n=1..100.

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

FORMULA

a(n^2) = n for all n>=1, a(n^2-4+1) = n for all n>=3. - Corrected by Peter Munn, Feb 24 2018

For n > 23, if A068527(n) = 1, 2, 4, 5, 7, 10 or 13 then a(n) = ceiling(sqrt(n)) + 1 else a(n) = ceiling(sqrt(n)). - Peter Munn, Feb 23 2018

EXAMPLE

            a(9)=3               a(10)=4                a(11)=5

              *                     *                      *

             / \                   / \                    / \

            *---*                 *---*                  +   +

           / \ / \               / \ / \                /     \

          *---*---*             *---*---*              +       +

         / \ / \ / \           / \ / \ / \            /         \

        *---*---*---*         +   *---*   +          *---+---+---*

                             /     \ /     \        / \ / \     / \

                            *---+---*---+---*      *---*---*   +   +

                                                  / \ / \ / \ /     \

                                                 *---*---*---*---+---*

.

           a(12)=6                a(13)=4                a(14)=5

              *                      *                      *

             / \                    / \                    / \

            *---*                  *---*                  +   +

           / \ / \                / \ / \                /     \

          *---*---*              *---*---*              +       +

         / \ / \ / \            / \ / \ / \            /         \

        *---*---*---*          *---*   *---*          *---+---+---*

       / \         / \        / \ /     \ / \        / \ / \ / \ / \

      *   +       +   +      *---*---*---*---*      *---*---*---*   +

     /     \     /     \                           / \ / \ / \ /     \

    +       +   +       +                         *---*---*---*---+---*

   /         \ /         \

  *---+---+---*---+---+---*

.

           a(15)=6                 a(16)=4                a(17)=5

              *                       *                      *

             / \                     / \                    / \

            +   +                   *---*                  +   +

           /     \                 / \ / \                /     \

          +       +               *---*---*              +       +

         /         \             / \ / \ / \            /         \

        +           +           *---*---*---*          *---*---*---*

       /             \         / \ / \ / \ / \        / \ / \ / \ / \

      *---*---*---*---*       *---*---*---*---*      *---*---*---*---*

     / \     / \     / \                            / \ / \ / \ / \ / \

    *---*   *---*   *---*                          *---*---*---*---*---*

   / \ / \ / \ / \ / \ / \

  *---*---*---*---*---*---*

.

           a(18)=6                 a(19)=5                 a(20)=6

              *                       *                       *

             / \                     / \                     / \

            +   +                   +   +                   *---*

           /     \                 /     \                 / \ / \

          +       +               *---*---*               *---*---*

         /         \             / \     / \             / \ / \ / \

        +           +           *---*   *---*           *---*---*---*

       /             \         / \ / \ / \ / \         / \ / \ / \ / \

      *---*---*---*---*       *---*---*---*---*       +   *---*---*   +

     / \ / \ / \ / \ / \     / \ / \ / \ / \ / \     /     \ / \ /     \

    *---*---*   *---*---*   *---*---*---*---*---*   +       *---*       +

   / \ / \ /     \ / \ / \                         /         \ /         \

  *---*---*---+---*---*---*                       *---+---+---*---+---+---*

CROSSREFS

Cf. A068527, A123120, A290820, A299705.

Sequence in context: A274441 A213859 A101336 * A137218 A087819 A290820

Adjacent sequences:  A299998 A299999 A300000 * A300002 A300003 A300004

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Feb 20 2018

EXTENSIONS

a(21)-a(100) from Peter Munn, Feb 24 2018

STATUS

approved

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Last modified November 14 08:13 EST 2018. Contains 317174 sequences. (Running on oeis4.)