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%I #24 Jan 30 2015 11:11:47
%S 1,1,7,257,50128,50796983,264719566561,7063448084710944,
%T 963204439792722969647,670733745303300958404439297,
%U 2384351527902618144856749327661056,43263422878945294225852497665519673400479,4006622856873663241294794301627790673728956619649
%N 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.
%C 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.)
%H Steve Butler, <a href="/A254127/b254127.txt">Table of n, a(n) for n = 0..15</a>
%H Z. Zhang, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match56/n3/match56n3_625-636.pdf">Merrifield-Simmons index of generalized Aztec diamond and related graphs</a>, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.
%e a(2)=7 for the following 7 tilings:
%e _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e |_|_| |_ _| |_|_| | |_| |_| | |_ _| | | |
%e |_|_| |_|_| |_ _| |_|_| |_|_| |_ _| |_|_|
%o # SAGE
%o def matrix_entry(L1,L2,n):
%o ....tally=0
%o ....for i in range(n-1):
%o ........if (not i in L1) and (not i in L2) and (not i+1 in L1) and (not i+1 in L2):
%o ............tally+=1
%o ....return 2^tally
%o def a(n):
%o ....index_set={}
%o ....counter=0
%o ....for C in Combinations(n):
%o ........index_set[counter]=C
%o ........counter+=1
%o ....current_v=[0]*counter
%o ....current_v[0]=1
%o ....for t in range(n):
%o ........new_v=[0]*counter
%o ........for i in range(counter):
%o ............for j in range(counter):
%o ................new_v[i]+=current_v[j]*matrix_entry(index_set[i],index_set[j],n)
%o ........current_v=new_v
%o ....return current_v[0]
%Y Cf. A052961, A254124, A254125, A254126.
%Y Main diagonal of A254414.
%K nonn
%O 0,3
%A _Steve Butler_, Jan 25 2015
%E a(0)=1 prepended by _Alois P. Heinz_, Jan 30 2015
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