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

%I A112833 #22 May 19 2020 06:00:06
%S A112833 1,2,5,20,117,1024,13357,259920,7539421,326177280,21040987113,
%T A112833 2024032315968,290333133984905,62102074862600192,19808204598680574457,
%U A112833 9421371079480456587520,6682097668647718038428569,7067102111711681259234263040,11145503882824383823706372042925
%N A112833 Number of domino tilings of a 3-pillow of order n.
%C A112833 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.
%C A112833 a(n)^(1/n^2) tends to 1.2211384384439007690866503099... - _Vaclav Kotesovec_, May 19 2020
%H A112833 Alois P. Heinz, <a href="/A112833/b112833.txt">Table of n, a(n) for n = 0..100</a>
%H A112833 C. Hanusa, <a href="http://people.qc.cuny.edu/faculty/christopher.hanusa/research/Documents/papers/Dissertation.pdf">A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows</a>, PhD Thesis, 2005, University of Washington, Seattle, USA.
%e A112833 The number of domino tilings of the 3-pillow of order 4 is 117=3^2*13.
%p A112833 with(LinearAlgebra):
%p A112833 b:= proc(x, y, k) option remember;
%p A112833       `if`(y>x or y<x/2, 0, `if`(x=k, `if`(y=k, 1, 0),
%p A112833          b(x, y-1, k)+b(x-1, y, k)+b(x-1, y-1, k)))
%p A112833     end:
%p A112833 P:= n-> Matrix(n, (i, j)-> b(i-1, i-1, j-1)):
%p A112833 R:= n-> Matrix(n, (i, j)-> `if`(i+j=n+1, 1, 0)):
%p A112833 a:= n-> Determinant(P(n)+R(n).(P(n)^(-1)).R(n)):
%p A112833 seq(a(n), n=0..20);  # _Alois P. Heinz_, Apr 26 2013
%t A112833 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_ *)
%Y A112833 This sequence breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.
%Y A112833 5-pillows: A112836-A112838; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.
%Y A112833 Related to A071101 and A071100.
%K A112833 nonn
%O A112833 0,2
%A A112833 Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005