A290697 -id:A290697 - 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

A290821 Side length of largest equilateral triangle that can be made from n or fewer equilateral triangles with integer sides s_k, subject to gcd(s_1,s_2,...,s_n) = 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 39, 49

Offset: 1

Hugo Pfoertner, Aug 11 2017

No construction from 2, 3 or 5 equilateral triangles exists. The first difference from the Padovan numbers occurs for a(15)=39, where the corresponding term A000931(19)=37. a(16)=A000931(20)=49. a(n) >= A000931(n+3). From the growth behavior of A290697 it is conjectured that a(k) > A000931(k+3) for all k > 20.
a(19) is at least 130. This compares with A000931(23) = 114. It hints of growth behavior similar to sqrt(A014529) or sqrt(A001590). Ceiling(sqrt(A001590(n))) matches a(n) to n=14, then runs 38, 52, 70, 95, 128, ... . - Peter Munn, Mar 10 2018
From Peter Munn, Mar 14 2018 re monotonicity: (Start)
For n >= 6, a(n+1) > a(n).
Sketch of proof (inductive step) expressed in terms of tiling:
Given a triangle of side a(n) tiled with n equilateral triangular tiles. Let X, Y and Z be the tiles incident on its vertices, with X being not smaller than Y or Z.
Case 1: Y and Z have no vertices coincident. Remove Y and Z, thereby reducing the tiled area to a pentagon that has edges A and C that were previously internal to the area, and an edge B between A and C. Fit a new tile T against edge B, thereby extending edges A and C. Make the tiled area triangular by fitting a new tile against each of the extended edges.
Case 2: X, Y and Z have pairwise coincident vertices. It follows that these tiles are the same size. Remove Y and Z, thereby reducing the tiled area to a rhombus. Remove the tile at the rhombus vertex opposite X. The remaining area is a pentagon, since n >= 6. Extend the area by resiting Y against X, and Z against Y so that X and Z have external edges aligned. Make the area trapezoidal by fitting a new tile against the area's edge that includes an edge of Y. Fit another tile T against the smaller of the trapezoid's parallel edges.
In each case, we now have n+1 tiles, tiling an equilateral triangle with side length a(n) plus the side of T. As the sides of new and removed tiles can be calculated by adding sides of tiles that stayed in place, the GCD of the sides is unchanged.
(End)

			a(12) = 16:
                                  *
                                 / \
                                +   +
                               /     \
                              +       +
                             /         \
                            +           +
                           /             \
                          +               +
                         /                 \
                        +                   +
                       /                     \
                      +                       +
                     /                         \
                    +                           +
                   /                             \
                  +                               +
                 /                                 \
                *---+---*---+---+---+---+---+---+---*
               / \     / \                         / \
              +   +   +   +                       +   +
             /     \ /     \                     /     \
            *---*---*       +                   +       +
           / \ / \ /         \                 /         \
          +   *---*---+---+---*               +           +
         /     \             / \             /             \
        +       +           +   +           +               +
       /         \         /     \         /                 \
      +           +       +       +       +                   +
     /             \     /         \     /                     \
    +               +   +           +   +                       +
   /                 \ /             \ /                         \
  *---+---+---+---+---*---+---+---+---*---+---+---+---+---+---+---*

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 of a(15)=39.

Cf. A000931, A001590, A167123, A290653, A290697, A290820.
A014529 gives greatest area of any convex polygon constructable from such triangles.
A089047 is this sequence's equivalent for squares.

Definition modified and 5 terms prepended by Peter Munn, Mar 14 2018

A299705 Number of ways to dissect an equilateral triangle into n non-overlapping equilateral triangles counting isomorphisms only once.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 2, 3, 9, 23, 62, 188, 574, 1826, 5953, 19664, 66049, 224700, 771859, 2674753

Offset: 1

Peter Munn and Hugo Pfoertner, Feb 17 2018

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

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

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.

Cf. A167123, A290697, A290821.

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

A290653 Number of perfect dissections of equilateral triangles into n equilateral triangles with integer sides.

Original entry on oeis.org

2, 2, 6, 23, 64, 181

Offset: 15

Hugo Pfoertner, Aug 08 2017

A perfect dissection has no two triangles of the same side. Triangles of different orientation are counted separately. The computation by A. Drapal and C. Hamalainen proves Tutte's conjecture that the smallest perfect dissection has size 15.

Ales Drapal and Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.
W. T. Tutte, The dissection of equilateral triangles into equilateral triangles, Mathematical Proceedings of the Cambridge Philosophical Society, 44(4), 463-482. doi:10.1017/S030500410002449X
Stuart Anderson, An Introduction to Triangled Equilateral Triangles.
Stuart Anderson, Illustration of dissections for n=15.
Stuart Anderson, Illustration of dissections for n=16.
Stuart Anderson, Illustration of dissections for n=17.
Stuart Anderson, Illustration of dissections for n=18.
Stuart Anderson, Illustration of dissections for n=19.
Stuart Anderson, Illustration of dissections for n=20.

Cf. A167123, A290697.

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.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Aug 11 2017

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

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

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.

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

Title changed as suggested by Peter Munn, Feb 17 2018

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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A290821 Side length of largest equilateral triangle that can be made from n or fewer equilateral triangles with integer sides s_k, subject to gcd(s_1,s_2,...,s_n) = 1.

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A299705 Number of ways to dissect an equilateral triangle into n non-overlapping equilateral triangles counting isomorphisms only once.

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A290653 Number of perfect dissections of equilateral triangles into n equilateral triangles with integer sides.

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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.

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Extensions