A290821 -id:A290821 - cp's OEIS frontend

A089047 Edge length of largest square dissectable into up to n squares in Mrs. Perkins's quilt problem.

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 13, 17, 23, 29, 41, 53, 70, 91, 126, 158, 216, 276, 386, 488, 675, 866, 1179, 1544, 2136, 2739, 3755, 4988, 6443

Offset: 1

R. K. Guy, Dec 03 2003

An inverse to A005670.
More precisely, a(n) = greatest k such that A005670(k) <= n. - Peter Munn, Mar 13 2018
It is not clear which terms have been proved to be correct and which are just conjectures. - Geoffrey H. Morley, Sep 07 2012; N. J. A. Sloane, Jul 06 2017
Terms up to and including a(18) have been proved correct by Ed Wynn (2013). - Stuart E Anderson, Sep 16 2013
A089046 and A089047 are almost certainly correct up to 5000. - Ed Pegg Jr, Jul 06 2017
Deleted terms above 5000. - N. J. A. Sloane, Jul 06 2017
Further best known terms are 8568, 11357, 14877, 19594, 26697, 34632. - Ed Pegg Jr, Jul 06 2017
A290821 is the equivalent sequence for equilateral triangles. - Peter Munn, Mar 06 2018

Ed Pegg, Jr., Mrs. Perkins's Quilts
Ed Pegg Jr. and Richard K. Guy, Mrs. Perkins's Quilts (Wolfram Demonstrations Project)
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420 [math.CO], 2013-2014.

Cf. A005670, A014529, A018835, A089046, A211302, A290821.

More terms from Ed Pegg Jr, Dec 03 2003
Corrected and extended by Ed Pegg Jr, Apr 18 2010
Duplicate a(6) deleted and a(22)-a(26) revised (from Ed Pegg Jr, Jun 15 2010) by Geoffrey H. Morley, Sep 07 2012
Conjectured terms have been extended up to a(44), based on simple squared square enumeration, by Duijvestijn, Skinner, Anderson, Pegg, Johnson, Milla and Williams. - Stuart E Anderson, Sep 16 2013
a(33) and further terms added by Ed Pegg Jr, Jul 06 2017
Name edited 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

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

A338861 a(n) is the largest area of a rectangle which can be dissected into n squares with integer sides s_i, i = 1 .. n, and gcd(s_1,...,s_n) = 1.

Original entry on oeis.org

1, 2, 6, 15, 42, 143, 399, 1190, 4209, 10920, 37245, 109886, 339745, 1037186, 3205734, 9784263, 29837784, 93313919, 289627536

Offset: 1

Rainer Rosenthal, Nov 12 2020

A219158 gives the minimum number of squares to tile an i x j rectangle. a(n) is found by checking all rectangles (i,j) for which A219158 has a dissection into n squares.

Due to the potential counterexamples to the minimal squaring conjecture (see MathOverflow link), terms after a(19) have to be considered only as lower bounds: a(20) >= 876696755, a(21) >= 2735106696. - Hugo Pfoertner, Nov 17 2020, Apr 02 2021

			a(6) = 11*13 = 143.
Dissection of the 11 X 13 rectangle into 6 squares:
.
          +-----------+-------------+
          |           |             |
          |           |             |
          |   6 X 6   |    7 X 7    |
          |           |             |
          |           |             |
          +---------+-+             |
          |         +-+-----+-------+
          |  5 X 5  |       |       |
          |         | 4 X 4 | 4 X 4 |
          |         |       |       |
          +---------+-------+-------+
.
a(19) = 16976*17061 = 289627536.
Dissection of the 16976 X 17061 rectangle into 19 squares:
.
       +----------------+-------------+
       |                |             |
       |                |             |
       |                |     7849    |
       |      9212      |             |
       |                |             |
       |                |             |
       |                |------+------|
       |________________|      |      |
       |             |   see   | 4109 |
       |             |Rosenthal|      |
       |             |  link +-+------+
       |     7764    |-------|        |
       |             |       |  5018  |
       |             | 4279  |        |
       |             |       |        |
       +-------------+-------+--------+
.

Stuart Anderson, Catalogues of Simple Perfect Squared Rectangles.
Stuart Anderson, Simple Imperfect Squared Rectangles, orders 9 to 24.
Bertram Felgenhauer, Filling rectangles with integer-sided squares.
MathOverflow, tiling a rectangle with the smallest number of squares, answer by Ed Pegg Jr, Jul 09 2017.
Rainer Rosenthal, Rectangle tiled by 19 squares with maximum area a(19)

Cf. A219158, A340726, A340920.

This sequence and A089047 are effectively analogs for dissecting (or tiling) rectangles and squares respectively. Analogs using equilateral triangular tiles are A014529 and A290821 respectively.

a(11)-a(17) from Hugo Pfoertner based on data from squaring.net website, Nov 17 2020
a(18) from Hugo Pfoertner, Feb 18 2021
a(19) from Hugo Pfoertner, Apr 02 2021

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

cp's OEIS Frontend

A089047 Edge length of largest square dissectable into up to n squares in Mrs. Perkins's quilt problem.

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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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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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A338861 a(n) is the largest area of a rectangle which can be dissected into n squares with integer sides s_i, i = 1 .. n, and gcd(s_1,...,s_n) = 1.

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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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