A056840 -id:A056840 - cp's OEIS frontend

A030222 Number of n-polyplets (polyominoes connected at edges or corners); may contain holes.

1, 2, 5, 22, 94, 524, 3031, 18770, 118133, 758381, 4915652, 32149296, 211637205, 1401194463, 9321454604, 62272330564, 417546684096

Offset: 1

Matthew Cook

See A056840 for illustrations, valid also for this sequence up to n=4, but slightly misleading for polyplets with holes. See the colored areas in the illustration of A056840(5)=99 which correspond to identical 5-polyplets. (The 2+2+4-3 = 5 additional figures counted there correspond to the 4-square configuration with a hole inside ({2,4,6,8} on a numeric keyboard), with one additional square added in three inequivalent places: "inside" one angle (touching two sides), attached to one side, and attached to a corner. These do only count for 3 here, but for 8 in A056840.) It can be seen that A056840 counts a sort of "spanning trees" instead, i.e., simply connected graphs that connect all of the vertices (using only "King's moves", and maybe other additional constraints). - M. F. Hasler, Sep 29 2014

			XXX..XX...XX..X.X..X.. (the 5 for n=3)
.......X...X...X....X.
.....................X

M. F. Hasler, Illustration of A030222(5)=94 through a colored version of Vicher's image for A056840(5)=99. (Figures filled with same color do not count as different here.)
Eric Weisstein's World of Mathematics, Polyplet.
Wikipedia, Pseudo-polyomino

Cf. A006770.

10th row of A366766.

Computed by Matthew Cook; extended by David W. Wilson

More terms from Joseph Myers, Sep 26 2002

A056841 Number of diagonal polyominoes with n cells.

Original entry on oeis.org

1, 1, 2, 5, 15, 54, 212, 908, 4011, 18260, 84320, 394462, 1860872, 8843896, 42275308, 203113670, 980101070, 4747504560, 23074132601

Offset: 1

James Sellers, Aug 28 2000

Apparently the cells are circular blobs which must be connected diagonally and the polyominoes can be rotated by 90 degrees and turned over.
Also the number of essentially different (i.e., not related by reflections, translations or rotations) diagrams consisting of n nodes in Z^2 and n-1 horizontal or vertical edges of length 1 between pairs of nodes such that the resulting graph is connected (hence a tree). - Paul Boddington, Jul 27 2004
They are thus equivalent to a subset of the polyedges, counted by A019988, i.e., those that are treelike. - John Mason, Aug 20 2021
The number of treelike polyedges with n edges is a(n+1). - John Mason, Feb 12 2023

			The diagonal polyominoes with 1, 2, 3 and 4 cells are
  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

R. J. Mathar, Table of all such polyominoes with n <= 10 cells (gzipped)
R. J. Mathar, C++ program
Douglas A. Torrance, Enumeration of planar Tangles, arXiv:1906.01541 [math.CO], 2019-2020. See Table 4.1 (C).
M. Vicher, Polyforms
M. Vicher, The 15 5-celled diagonal polyominoes
M. Vicher, The 15 5-celled diagonal polyominoes

See also A056840, A056787, A019988 (free polysticks), A348095 (with cycles).

a(n+1) + A348095(n) = A019988(n). - R. J. Mathar, Sep 30 2021

Description revised by N. J. A. Sloane, Jun 21 2001
a(10) from R. J. Mathar, Apr 10 2006
a(11) from Douglas A. Torrance, Mar 06 2020
a(12)-a(14) from John Mason, Aug 14 2021
a(15)-a(19) from John Mason, Jun 01 2023

A056787 Number of incongruental unlabeled undirected trees with n nodes on a square lattice and edges of length 1 or sqrt(2) admitted to the 4 nearest or 4 2nd nearest neighbors.

Original entry on oeis.org

1, 2, 6, 40, 303, 2929, 29752, 316935

Offset: 1

James Sellers, Aug 28 2000

a(n)>=A056841(n) since the trees of A056841 are a subset of these here. Edges along diagonals may cross.

			For n=2 we have
o-o
and
..o
./.
o..
as the only a(2)=2 candidates. Trees contributing to n=7 are
o.o-o
|\.\.
o.o-o
...\.
....o
or
o....
|\...
o.o-o
...X.
..o.o
./...
o....
where dashes are edges in E, NE, N, NW, W, SW, S or SE direction that connect nodes marked 'o' horizontally, vertically or along diagonals, and X's are crossing diagonal edges.

M. Vicher, Polyforms
R. J. Mathar, C++ program
R. J. Mathar, Polyforms (ASCII art)

A362223 a(n) is the number of n-celled fixed rounded polyominoes.

Original entry on oeis.org

1, 4, 20, 113, 682, 4294, 27830, 184240, 1239575, 8446660, 58150684, 403735761

Offset: 1

John Mason, Apr 11 2023

See A056840 for the definition of rounded polyominoes.

Cf. A056840, A362224.

A362224 a(n) is the number of n-celled one-sided rounded polyominoes.

Original entry on oeis.org

1, 2, 6, 34, 177, 1105, 6992, 46264, 310126, 2113012, 14539215, 100943151

Offset: 1

John Mason, Apr 11 2023

See A056840 for the definition of rounded polyominoes.

Cf. A056840, A362223.

cp's OEIS Frontend

A030222 Number of n-polyplets (polyominoes connected at edges or corners); may contain holes.

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A056841 Number of diagonal polyominoes with n cells.

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A056787 Number of incongruental unlabeled undirected trees with n nodes on a square lattice and edges of length 1 or sqrt(2) admitted to the 4 nearest or 4 2nd nearest neighbors.

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A362223 a(n) is the number of n-celled fixed rounded polyominoes.

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A362224 a(n) is the number of n-celled one-sided rounded polyominoes.

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