A112833 Number of domino tilings of a 3-pillow of order n.

1, 2, 5, 20, 117, 1024, 13357, 259920, 7539421, 326177280, 21040987113, 2024032315968, 290333133984905, 62102074862600192, 19808204598680574457, 9421371079480456587520, 6682097668647718038428569, 7067102111711681259234263040, 11145503882824383823706372042925

Offset

0,2

Comments

A 3-pillow is also called an Aztec pillow. The 3-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 3 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

a(n)^(1/n^2) tends to 1.2211384384439007690866503099... - Vaclav Kotesovec, May 19 2020

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

C. Hanusa, A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows, PhD Thesis, 2005, University of Washington, Seattle, USA.

Example

The number of domino tilings of the 3-pillow of order 4 is 117=3^2*13.

Maple

with(LinearAlgebra):
b:= proc(x, y, k) option remember;
      `if`(y>x or y<x/2, 0, `if`(x=k, `if`(y=k, 1, 0),
         b(x, y-1, k)+b(x-1, y, k)+b(x-1, y-1, k)))
    end:
P:= n-> Matrix(n, (i, j)-> b(i-1, i-1, j-1)):
R:= n-> Matrix(n, (i, j)-> `if`(i+j=n+1, 1, 0)):
a:= n-> Determinant(P(n)+R(n).(P(n)^(-1)).R(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 26 2013

Mathematica

b[x_, y_, k_] := b[x, y, k] = If[y>x || y<x/2, 0, If[x == k, If[y == k, 1, 0], b[x, y-1, k] + b[x-1, y, k] + b[x-1, y-1, k]]]; P[n_] := Table[ b[i-1, i-1, j-1], {i, 1, n}, {j, 1, n}]; R[n_] := Table[If[i+j == n+1, 1, 0], {i, 1, n}, {j, 1, n}]; a[0] = 1; a[n_] := Det[ P[n] + R[n].Inverse[P[n]].R[n]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 08 2015, after Alois P. Heinz *)

Crossrefs

This sequence breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.

5-pillows: A112836-A112838; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.

Related to A071101 and A071100.

Sequence in context: A000130 A288841 A009599 * A144503 A012321 A012519

Adjacent sequences: A112830 A112831 A112832 * A112834 A112835 A112836

Keyword

nonn

Author

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

Status

approved