A254127 The number of tilings of an n X n rectangle using integer length rectangles with at least one side of length 1, i.e., tiles are of size (1 X i) or (i X 1) with 1<=i<=n.

1, 1, 7, 257, 50128, 50796983, 264719566561, 7063448084710944, 963204439792722969647, 670733745303300958404439297, 2384351527902618144856749327661056, 43263422878945294225852497665519673400479, 4006622856873663241294794301627790673728956619649

Offset

0,3

Comments

Let R(n) be the set of squares that have vertices at integer coordinates and lie in the region of the plane |x|+|y|<=n+1, and let two squares be independent if they do not share a common edge. Then a(n) is the number of ways to pick a set of independent cell(s) in R(n). (Note R(n) is also known as the Aztec diamond.)

Links

Steve Butler, Table of n, a(n) for n = 0..15

Z. Zhang, Merrifield-Simmons index of generalized Aztec diamond and related graphs, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.

Example

a(2)=7 for the following 7 tilings:
   _ _   _ _   _ _   _ _   _ _   _ _   _ _
  |_|_| |_ _| |_|_| | |_| |_| | |_ _| | | |
  |_|_| |_|_| |_ _| |_|_| |_|_| |_ _| |_|_|

Prog

# SAGE
def matrix_entry(L1, L2, n):
....tally=0
....for i in range(n-1):
........if (not i in L1) and (not i in L2) and (not i+1 in L1) and (not i+1 in L2):
............tally+=1
....return 2^tally
def a(n):
....index_set={}
....counter=0
....for C in Combinations(n):
........index_set[counter]=C
........counter+=1
....current_v=[0]*counter
....current_v[0]=1
....for t in range(n):
........new_v=[0]*counter
........for i in range(counter):
............for j in range(counter):
................new_v[i]+=current_v[j]*matrix_entry(index_set[i], index_set[j], n)
........current_v=new_v
....return current_v[0]

Crossrefs

Cf. A052961, A254124, A254125, A254126.

Main diagonal of A254414.

Sequence in context: A188421 A165437 A232304 * A203968 A174251 A269576

Adjacent sequences: A254124 A254125 A254126 * A254128 A254129 A254130

Keyword

nonn

Author

Steve Butler, Jan 25 2015

Extensions

a(0)=1 prepended by Alois P. Heinz, Jan 30 2015

Status

approved