A056841 Number of diagonal polyominoes with n cells.

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

Offset: 1

Author

James Sellers, Aug 28 2000

Keywords

• nonn
• nice
• more

Comments

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

Examples

			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
		

Links

• 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

Crossrefs

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

Formula

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

Extensions

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