A290820 -id:A290820 - cp's OEIS frontend

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.

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

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

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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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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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