A299705 -id:A299705 - cp's OEIS frontend

A167123 Number of isomorphism classes of separated dissections of an equilateral triangle into n nonoverlapping equilateral triangles.

1, 0, 0, 1, 0, 1, 2, 3, 8, 20, 55, 161, 478, 1496, 4804, 15589, 51377, 172162, 583810, 1998407

Offset: 1

Jonathan Vos Post, Oct 27 2009

nonn

A dissection into 5 triangles is impossible.
From table on p.11 of Drapal. The authors write: We enumerate all dissections of an equilateral triangle into smaller equilateral triangles. We confirm W. T. Tutte's claim that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20.
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

			a(8)=3:
            *
           / \
          /   \
         /     \
        /       \
       /         \
      *---*-------*
     / \ / \     / \
    *---*   \   /   \
   / \ /     \ /     \
  *---*-------*-------*
            *
           / \
          /   \
         /     \
        *-------*
       / \     / \
      *---*   /   \
     / \ / \ /     \
    /   *---*       \
   /     \ /         \
  *-------*-----------*
          *
         / \
        *---*
       / \ / \
      *---*   \
     / \ /     \
    *---*       \
   / \ /         \
  *---*-----------*

Ales Drapal, Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.
Stuart Anderson, An Introduction to Triangled Equilateral Triangles.

Cf. A290653, A290697, A299705.

a(1)-a(3) added and name accommodated as suggested by M. F. Hasler, Feb 23 2018
Corrected and extended by Hugo Pfoertner, Aug 09 2017
Definition corrected by Hugo Pfoertner, Feb 17 2018

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.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Feb 20 2018

nonn

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

			            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
              *                       *                       *
             / \                     / \                     / \
            +   +                   +   +                   *---*
           /     \                 /     \                 / \ / \
          +       +               *---*---*               *---*---*
         /         \             / \     / \             / \ / \ / \
        +           +           *---*   *---*           *---*---*---*
       /             \         / \ / \ / \ / \         / \ / \ / \ / \
      *---*---*---*---*       *---*---*---*---*       +   *---*---*   +
     / \ / \ / \ / \ / \     / \ / \ / \ / \ / \     /     \ / \ /     \
    *---*---*   *---*---*   *---*---*---*---*---*   +       *---*       +
   / \ / \ /     \ / \ / \                         /         \ /         \
  *---*---*---+---*---*---*                       *---+---+---*---+---+---*

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

Cf. A068527, A123120, A290820, A299705.

a(n^2) = n for all n>=1, a(n^2-3) = 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

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

A358715 a(n) is the number of distinct ways to cut an equilateral triangle with edges of size n into equilateral triangles with integer sides.

Original entry on oeis.org

1, 2, 5, 26, 220, 3622, 105859, 5677789, 553715341, 98404068313, 31850967186980, 18779046566454536, 20167518569123722322, 39451359692134386945019

Offset: 1

Craig Knecht and John Mason, Nov 28 2022

In other words, the number of equilateral triangular tilings of an equilateral triangle, where rotations and reflections are considered distinct.

			a(3)=5 because of:
    /\      /\      /\      /\      /\
   /  \    /\/\    /  \    /\/\    /\/\
  /    \  /  \/\  /\/\/\  /\/  \  /\/\/\

Cf. A045846, A097076, A358716.

Cf. A290820, A290821, A299705, A300001.

a(10)-a(14) from Walter Trump, Dec 03 2022

A358716 a(n) is the number of inequivalent ways to cut an equilateral triangle with edges of size n into equilateral triangles with integer sides.

Original entry on oeis.org

1, 2, 3, 12, 50, 711, 18031, 952013, 92323440

Offset: 1

Craig Knecht and John Mason, Nov 28 2022

Similar to A358715, but now we do not regard dissections which differ by a rotation and/or reflection as distinct.

			a(3)=3 because of:
    /\      /\      /\
   /  \    /\/\    /\/\
  /    \  /  \/\  /\/\/\

Cf. A224239, A358715.

Cf. A299705.

cp's OEIS Frontend

A167123 Number of isomorphism classes of separated dissections of an equilateral triangle into n nonoverlapping equilateral triangles.

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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.

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A358715 a(n) is the number of distinct ways to cut an equilateral triangle with edges of size n into equilateral triangles with integer sides.

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A358716 a(n) is the number of inequivalent ways to cut an equilateral triangle with edges of size n into equilateral triangles with integer sides.

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