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

0,4

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Andrew Howroyd, Formula and PARI program, 2023.

Wikipedia, Polyomino

Example

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.

Prog

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

Crossrefs

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

Cf. A126764 (the fixed case).

Sequence in context: A056180 A000076 A217730 * A347760 A218576 A054169

Adjacent sequences: A360052 A360053 A360054 * A360056 A360057 A360058

Keyword

nonn

Author

Allan C. Wechsler and John Mason, Jan 23 2023

Extensions

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

Status

approved