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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. 5
1, 1, 7, 257, 50128, 50796983, 264719566561, 7063448084710944, 963204439792722969647, 670733745303300958404439297, 2384351527902618144856749327661056, 43263422878945294225852497665519673400479, 4006622856873663241294794301627790673728956619649 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 15 18:38 EDT 2022. Contains 356148 sequences. (Running on oeis4.)