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A360631 Number of sets of integer-sided rectangular pieces that can tile a 2 X n rectangle.

%I A360631 #19 May 05 2025 15:48:35
%S A360631 1,2,4,10,22,44,91,172,326,595,1066,1849,3204,5365,8921,14581,23558,
%T A360631 37440,59127,91957,142060,217015,328939,493917,737249,1090432,1603439,
%U A360631 2341094,3398313,4900740,7032809,10031010,14241165,20112575,28276657,39566635,55140425,76499692,105731884,145550924
%N A360631 Number of sets of integer-sided rectangular pieces that can tile a 2 X n rectangle.
%H A360631 Robin Visser, <a href="/A360631/b360631.txt">Table of n, a(n) for n = 0..40</a>
%e A360631 From _Robin Visser_, May 01 2025: (Start)
%e A360631 For n = 1, there are a(1) = 2 possible sets of rectangular pieces that can tile a 2 x 1 rectangle: one 1 x 2 piece; or two 1 x 1 pieces.
%e A360631 For n = 2, there are a(2) = 4 possible sets of rectangular pieces that can tile a 2 x 2 rectangle: one 2 x 2 piece; two 1 x 2 pieces; one 1 x 2 piece and two 1 x 1 pieces; or four 1 x 1 pieces.
%e A360631 For n = 3, there are a(3) = 10 possible sets of rectangular pieces that can tile a 2 x 3 rectangle: one 2 x 3 piece; one 2 x 2 piece and one 1 x 2 piece; one 2 x 2 piece and two 1 x 1 pieces; two 1 x 3 pieces; one 1 x 3 piece, one 1 x 2 piece, and one 1 x 1 piece; one 1 x 3 piece and three 1 x 1 pieces; three 1 x 2 pieces; two 1 x 2 pieces and two 1 x 1 pieces; one 1 x 2 piece and four 1 x 1 pieces; or six 1 x 1 pieces. (End)
%o A360631 (Python)
%o A360631 def a(n):
%o A360631     A, B = [set() for i in range(n+1)], [set() for i in range(n+1)]
%o A360631     A[0].add(()); B[0].add(());
%o A360631     for (m,k) in [(x,y) for x in range(1, n+1) for y in range(1, x+1)]:
%o A360631         for p in A[m-k]: A[m].add(tuple(sorted(list(p)+[k])))
%o A360631         for (p, q) in [(x,y) for x in A[m] for y in A[m]]:
%o A360631             B[m].add(tuple(sorted([(1,c) for c in p]+[(1,c) for c in q])))
%o A360631         for p in B[m-k]: B[m].add(tuple(sorted(list(p)+[tuple(sorted((2,k)))])))
%o A360631     return len(B[n])  # _Robin Visser_, May 01 2025
%Y A360631 Second column of A360629.
%Y A360631 Cf. A000041 (1 x n rectangle), A360632 (3 x n rectangle).
%K A360631 nonn
%O A360631 0,2
%A A360631 _Pontus von Brömssen_, Feb 14 2023
%E A360631 More terms from _Robin Visser_, May 04 2025
%E A360631 a(0) = 1 prepended by _Robin Visser_, May 05 2025