A375861 -id:A375861 - 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

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).

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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