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A359020 Number of inequivalent tilings of a 4 X n rectangle using integer-sided square tiles.

%I A359020 #19 Mar 18 2023 11:35:13
%S A359020 1,1,4,6,13,39,115,295,861,2403,7048,20377,60008,175978,519589,
%T A359020 1532455,4531277,13395656,39639758,117301153,347248981,1028011708,
%U A359020 3043852214,9012879842,26689014028,79033362580,234045889421,693101137571,2052569508948
%N A359020 Number of inequivalent tilings of a 4 X n rectangle using integer-sided square tiles.
%H A359020 John Mason, <a href="/A359020/b359020.txt">Table of n, a(n) for n = 0..1000</a>
%H A359020 John Mason, <a href="/A359019/a359019.pdf">Counting free tilings of a rectangle</a>
%F A359020 For even n > 4
%F A359020 a(n) = (A054856(n) + compo(n) + 4 * A054856((n - 2) / 2) +
%F A359020 2 * A054856((n - 4) / 2) + 2 * A054856(n / 2) +
%F A359020 2 * Sum_{k=0..(n - 2) / 2} (A054856(k))) / 4
%F A359020 For odd n > 4
%F A359020 a(n) = (A054856(n) + compo(n) + 2 * A054856((n - 3) / 2) +
%F A359020 2 * A054856((n - 1) / 2) + 2 * Sum_ {k=0..(n - 3) / 2} (A054856(k))) / 4
%F A359020 Where compo(n) is the number of distinct compositions of n as a sum of 1, 2, (1+1) and 4.
%e A359020 a(3) is 6 because of:
%e A359020   +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
%e A359020   | | | | |     | |   | | |   | | |   | | | | | |
%e A359020   +-+-+-+ +     + +   +-+ +   +-+ +   +-+ +-+-+-+
%e A359020   | | | | |     | |   | | |   | | |   | | |   | |
%e A359020   +-+-+-+ +     + +-+-+-+ +-+-+-+ +-+-+-+ +   +-+
%e A359020   | | | | |     | |   | | | |   | | | | | |   | |
%e A359020   +-+-+-+ +-+-+-+ +   +-+ +-+   + +-+-+-+ +-+-+-+
%e A359020   | | | | |     | |   | | | |   | | | | | | | | |
%e A359020   +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
%Y A359020 Column k = 4 of A227690.
%Y A359020 Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
%Y A359020 Fixed: A000045, A002478, A054856, A054857, A219925, A219926, A219927, A219928, A219929.
%Y A359020 Free: A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.
%K A359020 nonn
%O A359020 0,3
%A A359020 _John Mason_, Dec 12 2022