Ed Wynn - cp's OEIS frontend

A364404 Number of (strictly) 1-connected cubic graphs on 2n nodes.

0, 0, 0, 0, 1, 4, 29, 186, 1435, 12671, 131820, 1590900, 21940512, 339723835

Offset: 1

Ed Wynn, Jul 22 2023

			For n=5, the unique 10-node cubic graph that is strictly 1-connected is:
   o     o
  /|\   /|\
 o-o o-o o-o
  \|/   \|/
   o     o

G. Brinkmann, J. Goedgebeur and B. D. McKay, snarkhunter.

Cf. A204199, A204198, A002851.

# The snarkhunter program (see Links) has an option "C2" for (at least) 2-connectivity. So a(n) is the difference between the outputs from "./snarkhunter X 3 ns" and "./snarkhunter X 3 ns C2", where X=2n.

A319159 Given an equilateral triangular grid with side n, containing n(n+1)/2 points, a(n) is the minimal number of points to be selected, such that any equilateral triangle of points will include at least one of the selection.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 22, 28, 35, 44, 53, 63, 74, 86

Offset: 1

Ed Wynn, Sep 12 2018

This is the complementary problem to A240114: a(n) + A240114(n) = n(n+1)/2.

This is the same problem as A227116 and A319158, except that here the triangles may have any orientation. Due to the additional requirements, a(n) >= A227116(n) >= A319158(n).

			For n=4, this sequence has the same value a(4)=4 as A227116 and A319158, but if we look at the three solutions to those sequences (unique up to symmetry), representing selected points by O:
        O             O             O
       O ,           . ,           . .
      , . O         , O .         . O .
     . O , .       O . , O       . O O .
We see that only the last of these is a solution here -- the others have rotated triangles not including any selected point (for example, as shown with commas).  The last selection is therefore the unique solution (up to symmetry) for a(4)=4.

Ed Wynn, A comparison of encodings for cardinality constraints in a SAT solver, arXiv:1810.12975 [cs.LO], 2018.

Cf. A227116, A240114, A319158.

A319158 Given an equilateral triangular grid with side n, containing n(n+1)/2 points, a(n) is the minimal number of points to be selected, such that any equilateral triangle of points will include at least one of the selection, if the triangle has the same orientation as the grid.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 13, 18, 23, 29, 35, 43, 51

Offset: 1

Ed Wynn, Sep 12 2018

This is the complementary problem to A157795: A157795(n-1) + a(n) = n(n+1)/2.

This is the same problem as A227116, except that here the triangles must have the same orientation as the grid. Here, the triangle's sides must be aligned with the sides of the grid, and the horizontal side of the triangle must be its base (assuming the grid has a horizontal base). A227116 is different in that it also includes upside-down triangles, rotated 180 degrees compared to the grid, since these have sides aligned with the grid (but different orientation).

			For n=5, there is a unique solution for a(5)=6 (representing selected points by O):
        O
       . .
      , O ,
     . O O .
    O . , . O
It can be seen that this is not a valid solution for A227116 because of the upside-down triangle of commas. One solution for A227116(5)=7 would be to select one of the commas as well.

Ed Wynn, A comparison of encodings for cardinality constraints in a SAT solver, arXiv:1810.12975 [cs.LO], 2018.

Cf. A157795, A227116, A319159.

A288425 Minimal number of vertices that must be selected from an n X n square grid so that any square of 4 vertices, regardless of orientation, will include at least one selected vertex.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 22, 30, 39, 50

Offset: 1

Ed Wynn, Jun 09 2017

See the formula and A240443 to deduce lower bounds here: for example, a(11) <= 63, a(12) <= 77.

			For n = 3, an extra selection is required compared to A152125 (which considers only squares with sides parallel to the grid), because of the angled square consisting of the midpoints of the edges. One solution (with selected points shown as X) is:
  X X .
  . X .
  . . .

Cf. A240443 (the complementary problem), A152125, A227116.

The number of squares to be considered is A002415.

a(n) = n^2 - A240443(n).

a(10) derived from A240443(10) by Hugo van der Sanden, Nov 04 2021

A240125 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has one reflective symmetry in an axis parallel to a side, but no other symmetries.

Original entry on oeis.org

0, 0, 0, 3, 5, 138, 201, 13032, 19990, 4095612, 7026883, 4451051502

Offset: 1

Ed Wynn, Apr 01 2014

			The three dissections for n=4, with the axis horizontal:
---------    ---------    ---------
|   | | |    |   | | |    | | | | |
|   -----    |   -----    ---------
|   | | |    |   |   |    |   | | |
---------    -----   |    |   -----
|   | | |    |   |   |    |   | | |
|   -----    |   -----    ---------
|   | | |    |   | | |    | | | | |
---------    ---------    ---------

Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A226980, A045846, A224239, A240123, A240124.

A240124 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has 180-degree rotational symmetry, but no other symmetries.

Original entry on oeis.org

0, 0, 0, 0, 2, 19, 109, 1781, 13660, 397689, 5368943, 289864745

Offset: 1

Ed Wynn, Apr 01 2014

			The two dissections for n=5:
-----------    -----------
|   |   | |    | |   | | |
|   |   ---    ---   -----
|   |   | |    | |   | | |
-----------    -----------
| | | | | |    | | | | | |
-----------    -----------
| |   |   |    | | |   | |
---   |   |    -----   ---
| |   |   |    | | |   | |
-----------    -----------

Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A226980, A045846, A224239, A240123, A240125.

A240123 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has a reflective symmetry in one diagonal, but no other symmetries.

Original entry on oeis.org

0, 0, 1, 3, 19, 107, 847, 8647, 119835, 2255123, 58125783, 2050662011

Offset: 1

Ed Wynn, Apr 01 2014

'Inequivalent' has the same sense as in A224239: we do not regard dissections that differ by a rotation and/or reflection as distinct.

			The three dissections for n=4:
---------    ---------    ---------
|   | | |    |   |   |    |     | |
|   -----    |   |   |    |     ---
|   | | |    |   |   |    |     | |
---------    ---------    |     ---
| | | | |    |   | | |    |     | |
---------    |   -----    ---------
| | | | |    |   | | |    | | | | |
---------    ---------    ---------

Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A226980, A045846, A224239, A240124, A240125.

A240122 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has 90-degree rotational symmetry and no reflective symmetry.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 12, 40, 154, 760, 3260, 22730

Offset: 1

Ed Wynn, Apr 01 2014

			The two dissections for n=6:
-------------    -------------
| |   | | | |    | |   | | | |
---   -------    ---   -------
| |   | |   |    | |   | |   |
---------   |    ---------   |
| | |   |   |    | | | | |   |
-----   -----    -------------
|   |   | | |    |   | | | | |
|   ---------        ---------
|   | |   | |    |   | |   | |
-------   ---    -------   ---
| | | |   | |    | | | |   | |
-------------    -------------

Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A226979, A045846, A224239, A240120, A240121.

A240121 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has two reflective symmetries in axes parallel to the sides, and no other reflective symmetries.

Original entry on oeis.org

0, 0, 0, 1, 0, 13, 5, 183, 75, 4408, 1501, 180324

Offset: 1

Ed Wynn, Apr 01 2014

The two reflective symmetries imply 180-degree (but not 90-degree) rotational symmetry.

			This dissection is the only example for n=4:
---------
| |   | |
---   ---
| |   | |
---------
| |   | |
---   ---
| |   | |
---------

Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A226979, A045846, A224239, A240120, A240122.

A240120 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has reflective symmetry in both diagonals and no other reflective symmetries.

Original entry on oeis.org

0, 0, 0, 1, 1, 9, 19, 121, 275, 2489, 7217, 86775

Offset: 1

Ed Wynn, Apr 01 2014

'Inequivalent' has the same sense as in A224239: we do not regard dissections that differ by a rotation and/or reflection as distinct.

The two reflective symmetries imply 180-degree (but not 90-degree) rotational symmetry.

			This is the single dissection for n=4:
---------
|   | | |
|   -----
|   | | |
---------
| | |   |
-----   |
| | |   |
---------

Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A226979, A045846, A224239, A240121, A240122.

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A364404 Number of (strictly) 1-connected cubic graphs on 2n nodes.

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A319159 Given an equilateral triangular grid with side n, containing n(n+1)/2 points, a(n) is the minimal number of points to be selected, such that any equilateral triangle of points will include at least one of the selection.

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A319158 Given an equilateral triangular grid with side n, containing n(n+1)/2 points, a(n) is the minimal number of points to be selected, such that any equilateral triangle of points will include at least one of the selection, if the triangle has the same orientation as the grid.

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A288425 Minimal number of vertices that must be selected from an n X n square grid so that any square of 4 vertices, regardless of orientation, will include at least one selected vertex.

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A240125 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has one reflective symmetry in an axis parallel to a side, but no other symmetries.

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A240124 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has 180-degree rotational symmetry, but no other symmetries.

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A240123 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has a reflective symmetry in one diagonal, but no other symmetries.

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A240122 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has 90-degree rotational symmetry and no reflective symmetry.

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A240121 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has two reflective symmetries in axes parallel to the sides, and no other reflective symmetries.

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A240120 Number of inequivalent ways to cut an n X n square into squares with integer sides, such that the dissection has reflective symmetry in both diagonals and no other reflective symmetries.

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