A375770 -id:A375770 - cp's OEIS frontend

A375860 In an n X n grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; a(n) is the number of solutions up to symmetries of the rectangle.

1, 2, 19, 298, 6351, 152516, 3982186, 112862604, 3477324887, 116564337340, 4243247421102, 167133260332116, 7090692457208714, 322501850458391072, 15654927194390330084, 807745569631252715576, 44138380646303320082391, 2546015709197766295214940, 154576479598451154017930998

Offset: 1

Andrew Howroyd, Aug 31 2024

nonn

See A375770 for additional information.

			a(3) = 19. The A375770(3) = 10 distinct solutions with their multiplicities are:
  ._._._.   ._._._.   ._._._.   ._._._.   ._._._.
  |     |   |   | |   |   | |   | | | |   |   ._|
  | | | |   | |   |   | | | |   |     |   | |   |
  |_|_|_|   |_|_|_|   |_|_._|   |_|_|_|   |_|_|_|
    (2)       (2)       (2)       (2)       (2)
  ._._._.   ._._._.   ._._._.   ._._._.   ._._._.
  | | ._|   |   | |   |   ._|   |_. ._|   |_. | |
  |     |   | | ._|   | | ._|   |     |   |   ._|
  |_|_|_|   |_|_._|   |_|_._|   |_|_|_|   |_|_._|
    (2)       (2)       (2)       (2)       (1)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Main diagonal of A375861.

Cf. A375770, A375817, A375859.

\\ See PARI link in A375770 for program code.
vector(20, n, A375860(n)) \\ Andrew Howroyd, Sep 03 2024

A375817 In an n X n grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; a(n) is the number of solutions not reduced for symmetries.

Original entry on oeis.org

1, 4, 56, 1112, 25000, 607712, 15918280, 451371888, 13908978792, 466254401360, 16972978214456, 668532916285104, 28362769354991656, 1290007395847848160, 62619708755213093360, 3230982278203826268640, 176553522584025285715304, 10184062836771923067636528

Offset: 1

Lars Blomberg, Aug 30 2024

nonn

This sequence contains some, but not all of the spanning trees in A007341, a(n)<A007341(n), for n>2.

See A375770 for examples.

			a(3) = 56. The A375770(3) = 10 distinct solutions with their multiplicities are:
  ._._._.   ._._._.   ._._._.   ._._._.   ._._._.
  |     |   |   | |   |   | |   | | | |   |   ._|
  | | | |   | |   |   | | | |   |     |   | |   |
  |_|_|_|   |_|_|_|   |_|_._|   |_|_|_|   |_|_|_|
    (4)       (8)       (4)       (2)       (8)
  ._._._.   ._._._.   ._._._.   ._._._.   ._._._.
  | | ._|   |   | |   |   ._|   |_. ._|   |_. | |
  |     |   | | ._|   | | ._|   |     |   |   ._|
  |_|_|_|   |_|_._|   |_|_._|   |_|_|_|   |_|_._|
    (8)       (8)       (8)       (4)       (2)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Main diagonal of A375858.

Cf. A007341, A375770 (up to rotations and reflections), A375859 (up to rotations), A375860 (up to symmetries of the rectangle).

\\ See PARI link in A375770 for program code.
vector(20, n, A375817(n)) \\ Andrew Howroyd, Sep 03 2024

Terms corrected and extended by Andrew Howroyd, Sep 03 2024

A375859 In an n X n grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; a(n) is the number of solutions up to rotations.

Original entry on oeis.org

1, 1, 17, 278, 6299, 151928, 3980838, 112842972, 3477284215, 116563600340, 4243245979546, 167133229071276, 7090692397872054, 322501848961962040, 15654927191595331100, 807745569550956567160, 44138380646154800057431, 2546015709192980766909132, 154576479598442363146107882

Offset: 1

Andrew Howroyd, Aug 31 2024

nonn

See A375770 and A375817 for additional information.

			a(3) = 17. The A375770(3) = 10 distinct solutions with their multiplicities are:
  ._._._.   ._._._.   ._._._.   ._._._.   ._._._.
  |     |   |   | |   |   | |   | | | |   |   ._|
  | | | |   | |   |   | | | |   |     |   | |   |
  |_|_|_|   |_|_|_|   |_|_._|   |_|_|_|   |_|_|_|
    (1)       (2)       (2)       (1)       (2)
  ._._._.   ._._._.   ._._._.   ._._._.   ._._._.
  | | ._|   |   | |   |   ._|   |_. ._|   |_. | |
  |     |   | | ._|   | | ._|   |     |   |   ._|
  |_|_|_|   |_|_._|   |_|_._|   |_|_|_|   |_|_._|
    (2)       (2)       (2)       (1)       (2)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A375770 (up to rotations and reflections), A375817 (not reduced for symmetry), A375860 (up to symmetries of rectangle).

\\ See PARI link in A375770 for program code.
vector(20, n, A375859(n)) \\ Andrew Howroyd, Sep 03 2024

A375858 Array T(n,m) read by antidiagonals: In an n X m grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; T(n,m) is the number of solutions.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 26, 56, 26, 1, 1, 57, 212, 212, 57, 1, 1, 120, 701, 1112, 701, 120, 1, 1, 247, 2179, 4793, 4793, 2179, 247, 1, 1, 502, 6600, 19082, 25000, 19082, 6600, 502, 1, 1, 1013, 19808, 74368, 116852, 116852, 74368, 19808, 1013, 1

Offset: 1

Andrew Howroyd, Aug 31 2024

See A375770 and A375817 for additional explanation and illustration of solutions.

This sequence counts a subset of the spanning trees enumerated in A116469.

			Array begins:
==================================================
n/m | 1   2    3     4      5       6        7 ...
----+---------------------------------------------
  1 | 1   1    1     1      1       1        1 ...
  2 | 1   4   11    26     57     120      247 ...
  3 | 1  11   56   212    701    2179     6600 ...
  4 | 1  26  212  1112   4793   19082    74368 ...
  5 | 1  57  701  4793  25000  116852   535776 ...
  6 | 1 120 2179 19082 116852  607712  3048668 ...
  7 | 1 247 6600 74368 535776 3048668 15918280 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Main diagonal is A375817.

Cf. A116469, A375770, A375859, A375860, A375861.

\\ A375858(n,m) defined in PARI link in A375770.

T(n,m) = T(m,n).

A375861 Array T(n,m) read by antidiagonals: In an n X m grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; T(n,m) is the number of solutions up to symmetries of the rectangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 9, 19, 9, 1, 1, 19, 63, 63, 19, 1, 1, 35, 192, 298, 192, 35, 1, 1, 71, 576, 1246, 1246, 576, 71, 1, 1, 135, 1698, 4857, 6351, 4857, 1698, 135, 1, 1, 271, 5042, 18768, 29467, 29467, 18768, 5042, 271, 1, 1, 527, 14963, 72968, 134397, 152516, 134397, 72968, 14963, 527, 1

Offset: 1

Andrew Howroyd, Aug 31 2024

See A375770 and A375860 for additional explanation and illustration of solutions.

			Array begins:
===============================================
n/m | 1  2    3     4      5      6       7 ...
----+------------------------------------------
  1 | 1  1    1     1      1      1       1 ...
  2 | 1  2    5     9     19     35      71 ...
  3 | 1  5   19    63    192    576    1698 ...
  4 | 1  9   63   298   1246   4857   18768 ...
  5 | 1 19  192  1246   6351  29467  134397 ...
  6 | 1 35  576  4857  29467 152516  763479 ...
  7 | 1 71 1698 18768 134397 763479 3982186 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Main diagonal is A375860.

Cf. A375858 (not reduced for symmetry).

\\ A375861(n,m) defined in PARI link in A375770.

T(n,m) = T(m,n).

cp's OEIS Frontend

A375860 In an n X n grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; a(n) is the number of solutions up to symmetries of the rectangle.

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A375817 In an n X n grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; a(n) is the number of solutions not reduced for symmetries.

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A375859 In an n X n grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; a(n) is the number of solutions up to rotations.

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A375858 Array T(n,m) read by antidiagonals: In an n X m grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; T(n,m) is the number of solutions.

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A375861 Array T(n,m) read by antidiagonals: In an n X m grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; T(n,m) is the number of solutions up to symmetries of the rectangle.

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Examples

Links

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Programs

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