A360055 Number of L-connected free polyominoes with n cells (see comments for definition).

1, 1, 1, 2, 4, 7, 14, 24, 48, 83, 155, 265, 472, 793, 1356, 2235, 3700, 5977, 9636, 15262, 24068, 37439, 57920, 88628, 134838, 203264, 304688, 453126, 670238, 984556, 1438878, 2089996, 3021162, 4343229, 6215668, 8851151, 12550775, 17716075, 24907961, 34873541, 48644474

Offset: 0

Author

Allan C. Wechsler and John Mason, Jan 23 2023

Keywords

• nonn

Comments

By "L-connected" we mean that every pair of cells in the polyomino can be connected by a chain of cells in the polyomino, consisting of a vertical chain and a horizontal chain that share one of their end cells.

We consider the empty polyomino, or "nullomino" to be L-connected because there are no pairs of cells, so the universal quantifier in the definition is trivially satisfied.

The entries through n = 6 were hand-counted by Allan C. Wechsler, and John Mason confirmed those entries and extended the data through n = 18.

Examples

			For n = 4 the 4 solutions are the straight tetromino, the L tetromino, the T tetromino, and the square tetromino. The skew or S tetromino is not L-connected, because there is no qualifying path between the two most distant cells.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Andrew Howroyd, Formula and PARI program, 2023.
• Wikipedia, Polyomino

Crossrefs

Main entry at A000105; see the cross-references there.

Cf. A126764 (the fixed case).

Programs

PARI

seq(40) \\ See Links - Andrew Howroyd, Jan 27 2023

Extensions

Terms a(19) and beyond from Andrew Howroyd, Jan 24 2023