Anna Tscharre - cp's OEIS frontend

User: Anna Tscharre

Anna Tscharre has authored 1 sequences.

A367252 a(n) is the number of ways to tile an n X n square as explained in comments.

1, 0, 1, 4, 88, 3939, 534560, 185986304, 175655853776, 437789918351688, 2898697572048432368, 50698981110982431863735, 2342038257118692026082013568, 285250169294740386915765591840768, 91531011920509198679773321121428857296, 77312253225939431362091700178995800855209496

Offset: 0

Anna Tscharre, Nov 11 2023

nonn

Draw a Dyck path from (0,0) to (n,n) so the path always stays above the diagonal. Now section the square into horizontal rows of height one to the left of the path and tile these rows using 1 X 2 and 1 X 1 tiles. Similarly, section the part to the right of the path into columns with width one and tile these using 2 X 1 and 1 X 1 tiles. Furthermore, no 1 X 1 tiles are allowed in the bottom row.

Anna Tscharre, A possible 6 X 6 tiling

Special case of A003150.

Cf. A055010, A307082.

b:= proc(x, y) option remember; (F->
     `if`(x=0 and y=0, 1, `if`(x>0, b(x-1, y)*F(y-1), 0)+
     `if`(y>x, b(x, y-1)*F(x+1), 0)))(combinat[fibonacci])
    end:
a:= n-> b(n$2):
seq(a(n), n=0..15);  # Alois P. Heinz, Nov 11 2023

b[x_, y_] := b[x, y] = With[{F = Fibonacci},
     If[x == 0 && y == 0, 1,
     If[x > 0, b[x - 1, y]*F[y - 1], 0] +
     If[y > x, b[x, y - 1]*F[x + 1], 0]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Nov 14 2023, after Alois P. Heinz *)

a(n) == 1 (mod 2) <=> n in { A055010 }. - Alois P. Heinz, Nov 11 2023

cp's OEIS Frontend

User: Anna Tscharre

A367252 a(n) is the number of ways to tile an n X n square as explained in comments.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula