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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. 4
1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 39, 49 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

No construction from 2, 3 or 5 equilateral triangles exists. The first difference to 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)

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 of a(15)=39.

EXAMPLE

a(12) = 16:

                                  *

                                 / \

                                +   +

                               /     \

                              +       +

                             /         \

                            +           +

                           /             \

                          +               +

                         /                 \

                        +                   +

                       /                     \

                      +                       +

                     /                         \

                    +                           +

                   /                             \

                  +                               +

                 /                                 \

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

               / \     / \                         / \

              +   +   +   +                       +   +

             /     \ /     \                     /     \

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

           / \ / \ /         \                 /         \

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

         /     \             / \             /             \

        +       +           +   +           +               +

       /         \         /     \         /                 \

      +           +       +       +       +                   +

     /             \     /         \     /                     \

    +               +   +           +   +                       +

   /                 \ /             \ /                         \

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

CROSSREFS

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.

Sequence in context: A228361 A182097 A290697 * A072493 A064324 A173090

Adjacent sequences:  A290818 A290819 A290820 * A290822 A290823 A290824

KEYWORD

nonn,hard,more

AUTHOR

Hugo Pfoertner, Aug 11 2017

EXTENSIONS

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

STATUS

approved

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Last modified August 18 16:13 EDT 2018. Contains 313833 sequences. (Running on oeis4.)