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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.

%I A254127 #35 Mar 04 2025 02:46:28
%S A254127 1,1,7,257,50128,50796983,264719566561,7063448084710944,
%T A254127 963204439792722969647,670733745303300958404439297,
%U A254127 2384351527902618144856749327661056,43263422878945294225852497665519673400479,4006622856873663241294794301627790673728956619649
%N 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.
%C A254127 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 A254127 Liang Kai, <a href="/A254127/b254127.txt">Table of n, a(n) for n = 0..27</a> (terms 0..15 from Steve Butler)
%H A254127 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 A254127 a(2)=7 for the following 7 tilings:
%e A254127    _ _   _ _   _ _   _ _   _ _   _ _   _ _
%e A254127   |_|_| |_ _| |_|_| | |_| |_| | |_ _| | | |
%e A254127   |_|_| |_|_| |_ _| |_|_| |_|_| |_ _| |_|_|
%o A254127 (SageMath)
%o A254127 def matrix_entry(L1, L2, n):
%o A254127     tally=0
%o A254127     for i in range(n-1):
%o A254127         if (not i in L1) and (not i in L2) and (not i+1 in L1) and (not i+1 in L2):
%o A254127             tally+=1
%o A254127     return 2^tally
%o A254127 def a(n):
%o A254127     index_set={}
%o A254127     counter=0
%o A254127     for C in Combinations(n):
%o A254127         index_set[counter]=C
%o A254127         counter+=1
%o A254127     current_v=[0]*counter
%o A254127     current_v[0]=1
%o A254127     for t in range(n):
%o A254127         new_v=[0]*counter
%o A254127         for i in range(counter):
%o A254127             for j in range(counter):
%o A254127                 new_v[i]+=current_v[j]*matrix_entry(index_set[I], index_set[j], n)
%o A254127         current_v=new_v
%o A254127     return current_v[0]
%o A254127 for n in range(0, 10):
%o A254127     print(a(n), end=', ')
%Y A254127 Cf. A052961, A254124, A254125, A254126.
%Y A254127 Main diagonal of A254414.
%K A254127 nonn
%O A254127 0,3
%A A254127 _Steve Butler_, Jan 25 2015
%E A254127 a(0)=1 prepended by _Alois P. Heinz_, Jan 30 2015