A345887 Number of tilings of an n-cell circular array with rectangular tiles of any size, and where the number of possible colors of a tile is given by the largest cell covered.

1, 6, 30, 164, 1030, 7422, 60620, 554248, 5611770, 62353010, 754471432, 9876716940, 139097096918, 2097156230470, 33704296561140, 575219994643472, 10389911153247730, 198019483156015578, 3971390745517868000, 83608226221428800020, 1843561388182505040462

Offset: 1

Author

Lara Pudwell, Jun 28 2021

Keywords

• nonn

Links

• Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.

Crossrefs

Cf. A002627, A193657.

Programs

Maple

a:= proc(n) a(n):= `if`(n=1, 1, a(n-1)*n^2/(n-1)+n) end:
seq(a(n), n=1..21);  # Alois P. Heinz, Jun 28 2021

Mathematica

With[{r = Range[21]}, r*Rest@ FoldList[Times @@ {##} + 1 &, 0, r]] (* Michael De Vlieger, Jun 28 2021 *)

PARI

a(n) = n*sum(k=1, n, n!/k!); \\ Michel Marcus, Jun 29 2021

Formula

a(n) = n * Sum_{k=1..n} n!/k!.

a(n) = n * A002627(n).

From Alois P. Heinz, Jun 28 2021: (Start)

E.g.f.: (exp(x)-x)/(x-1)^2 - exp(x).

a(n) = A193657(n) - 1. (End)

D-finite with recurrence a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2) -2 =0. - R. J. Mathar, Jan 11 2024