cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A375770 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 distinct under reflections and rotations.

Original entry on oeis.org

1, 1, 10, 149, 3177, 76258, 1991098, 56431302, 1738662461, 58282168670, 2121623710614, 83566630166058, 3545346228604588, 161250925229195536, 7827463597195165900, 403872784815626357788, 22069190323151660044413, 1273007854598883147607470, 77288239799225577008977654
Offset: 1

Views

Author

Lars Blomberg, Aug 27 2024

Keywords

Comments

This sequence contains some, but not all of the spanning trees in A349718.

Examples

			a(2)=1:
+=======+
| o - o |
| |   | |
| o ║ o |
+===+===+
a(3)=10:
+===========+  +=======+===+  +=======+===+  +===+===+===+  +===========+
| o - o - o |  | o - o ║ o |  | o - o ║ o |  | o ║ o ║ o |  | o - o - o |
| |   |   | |  | |   |   | |  | |   | ║ | |  | |   |   | |  | |   |   ══+
| o ║ o ║ o |  | o ║ o - o |  | o ║ o ║ o |  | o - o - o |  | o ║ o - o |
| | ║ | ║ | |  | | ║ |   | |  | | ║ |   | |  | |   |   | |  | | ║ |   | |
| o ║ o ║ o |  | o ║ o ║ o |  | o ║ o - o |  | o ║ o ║ o |  | o ║ o ║ o |
+===+===+===+  +===+===+===+  +===+=======+  +===+===+===+  +===+===+===+
+===+=======+  +=======+===+  +===========+  +===========+  +=======+===+
| o ║ o - o |  | o - o ║ o |  | o - o - o |  | o - o - o |  | o - o ║ o |
| |   |   ══+  | |   |   | |  | |   |   ══+  +═══  |   ══+  +═══  |   | |
| o - o - o |  | o ║ o - o |  | o ║ o - o |  | o - o - o |  | o - o - o |
| |   |   | |  | | ║ |   ══+  | | ║ |   ══+  | |   |   | |  | |   |   ══+
| o ║ o ║ o |  | o ║ o - o |  | o ║ o - o |  | o ║ o ║ o |  | o ║ o - o |
+===+===+===+  +===+=======+  +===+=======+  +===+===+===+  +===+=======+
n=4 sample
+===+===+===+===+  +=======+===+===+
| o ║ o ║ o ║ o |  | o - o ║ o ║ o |
| |   |   |   | |  +═══  | ║ |   | |
| o - o - o - o |  | o - o ║ o - o |
+═══  |   |   ══+  | |   | ║ |   ══+
| o - o ║ o - o |  | o ║ o ║ o - o |
| |   | ║ |   ══+  | | ║ |   |   | |
| o ║ o ║ o - o |  | o ║ o - o ║ o |
+===+===+=======+  +===+=======+===+
n=5 sample
+===+===+===+===+===+
| o ║ o ║ o ║ o ║ o |
| |   | ║ | ║ |   | |
| o - o ║ o ║ o - o |
| |   |   |   |   ══+
| o ║ o - o - o - o |
| | ║ |   |   |   ══+
| o ║ o ║ o ║ o - o |
| | ║ | ║ | ║ |   | |
| o ║ o ║ o ║ o ║ o |
+===+===+===+===+===+
n=6 sample
+===========+===+===+===+
| o - o - o ║ o ║ o ║ o |
| |   |   | ║ | ║ | ║ | |
| o ║ o ║ o ║ o ║ o ║ o |
| | ║ | ║ |   |   | ║ | |
| o ║ o ║ o - o - o ║ o |
| | ║ | ║ |   |   | ║ | |
| o ║ o ║ o ║ o ║ o ║ o |
| | ║ | ║ | ║ | ║ |   | |
| o ║ o ║ o ║ o ║ o - o |
| | ║ | ║ | ║ | ║ |   ══+
| o ║ o ║ o ║ o ║ o - o |
+===+===+===+===+=======+
Examples of spanning trees where some of the walls do not start at a border, so they are not included in this sequence.
+===+===+=======+  +===============+
| o ║ o ║ o - o |  | o - o - o - o |
| | ║ |   |   | |  +══════════   | |
| o ║ o - o ║ o |  | o - o - o ║ o |
| | ║ ═════ ║ | |  | |   ══  | ║ | |
| o ║ o - o ║ o |  | o ║ o - o ║ o |
| |   |   | ║ | |  | |   ═════   | |
| o - o ║ o - o |  | o - o - o - o |
+=======+=======+  +===============+

Links

Crossrefs

Cf. A349718, A375817 (not reduced for symmetries), A375859 (up to rotations), A375860 (up to symmetries of rectangle).

Programs

Extensions

a(1) set to 1 and a(9) onwards from Andrew Howroyd, Aug 31 2024

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.

Original entry on oeis.org

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

Views

Author

Andrew Howroyd, Aug 31 2024

Keywords

Comments

See A375770 for additional information.

Examples

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

Links

Crossrefs

Main diagonal of A375861.
Cf. A375770, A375817, A375859.

Programs

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

Views

Author

Andrew Howroyd, Aug 31 2024

Keywords

Comments

See A375770 and A375817 for additional information.

Examples

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

Links

Crossrefs

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

Programs

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

Views

Author

Andrew Howroyd, Aug 31 2024

Keywords

Comments

See A375770 and A375817 for additional explanation and illustration of solutions.
This sequence counts a subset of the spanning trees enumerated in A116469.

Examples

			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 ...
  ...

Links

Crossrefs

Main diagonal is A375817.
Cf. A116469, A375770, A375859, A375860, A375861.

Programs

Formula

T(n,m) = T(m,n).
Showing 1-4 of 4 results.

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