A387599 Number of ways to tile a strip of length n with squares, dominoes and trominoes, using an equal number of dominoes and trominoes.

1, 1, 1, 1, 1, 3, 7, 13, 21, 31, 49, 87, 163, 301, 531, 909, 1557, 2723, 4861, 8731, 15589, 27607, 48733, 86271, 153531, 274221, 490079, 874825, 1560103, 2782933, 4970479, 8889685, 15912333, 28489827, 51010617, 91349735, 163657705, 293360107, 526111273

Offset

0,6

Links

Table of n, a(n) for n=0..38.

Formula

a(n) = Sum_{i >= 0} binomial(n-3*i, 2*i) * binomial(2*i,i).

D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + (4*n-10)*a(n-5).

a(n) = Sum_{i=0..floor(n/5)} multinomial(n-3*i; i, i, n-5*i). - Alois P. Heinz, Sep 08 2025

a(n) = hypergeom([(1-n)/5, (2-n)/5, (3-n)/5, (4-n)/5, -n/5], [1, (1-n)/3, (2-n)/3, -n/3], 5^5/3^3). - Stefano Spezia, Sep 09 2025

G.f.: 1 / sqrt((1-x)^2 - 4*x^5). - Seiichi Manyama, Jan 18 2026

Example

For n=5, the a(5)=3 tilings are: all squares, domino-tromino, and tromino-domino.

Mathematica

Table[Sum[Binomial[n-3i, 2i]*Binomial[2i, i], {i, 0, n/5}], {n, 0, 38}]

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(1/sqrt((1-x)^2-4*x^5)) \\ Seiichi Manyama, Jan 18 2026

Crossrefs

Cf. A001850, A002426, A098479, A098482.

Sequence in context: A011898 A098577 A004136 * A335865 A147409 A147342

Adjacent sequences: A387596 A387597 A387598 * A387600 A387601 A387602

Keyword

nonn

Author

Greg Dresden and Anna Kalynchuk, Sep 02 2025

Status

approved