A359703 Number of fillomino dissections of a 2 X n rectangle.

1, 1, 5, 33, 138, 715, 3524, 17119, 84655, 416723, 2047650, 10072806, 49542408, 243701785, 1198732022, 5895900754, 28999718642, 142641530115, 701610208573, 3450988507136, 16974245195432, 83490673950264, 410663317558386, 2019918477187441, 9935315439670326

Offset

0,3

Comments

A fillomino dissection of a rectangle is a tiling by polyominoes in which no two polyominoes of the same size are adjacent.

The sequence a(n+1)/a(n) appears to converge rapidly to 4.91867 12250 37424 13083 06703 91572 28440 1... (with a baffling sequence of sign changes in a(n+2)a(n)-a(n+1)^2). - Don Knuth, Jan 15 2023

References

Donald E. Knuth, The Art of Computer Programming, exercise in Section 7.2.2.3 (in preparation).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..125

Don Knuth, fillomino2xn.m

Wikipedia, Fillomino

Example

For n=2 the a(2)=5 dissections are
  13 31 33 33 44
  33 33 13 31 44

Crossrefs

The fillomino dissections of a 1 X n rectangle are Carlitz compositions (sequence A003242).

Sequence in context: A211059 A292000 A146263 * A255943 A270222 A270279

Adjacent sequences: A359700 A359701 A359702 * A359704 A359705 A359706

Keyword

nonn

Author

Don Knuth, Jan 11 2023

Extensions

a(20)-a(24) from Alois P. Heinz, Jan 12 2023

Status

approved