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A359019 Number of inequivalent tilings of a 3 X n rectangle using integer-sided square tiles.

%I A359019 #22 Mar 18 2023 11:34:58
%S A359019 1,1,2,3,6,10,21,39,82,163,347,717,1533,3232,6927,14748,31645,67690,
%T A359019 145322,311535,668997,1435645,3083301,6619842,14218066,30533005,
%U A359019 65580338,140847132,302522253,649759735,1395611508,2997573501,6438470626,13829057884,29703388721,63799607283,137035047576,294336860797,632205714741
%N A359019 Number of inequivalent tilings of a 3 X n rectangle using integer-sided square tiles.
%H A359019 John Mason, <a href="/A359019/b359019.txt">Table of n, a(n) for n = 0..1000</a>
%H A359019 John Mason, <a href="/A359019/a359019_1.pdf">Counting free tilings of a rectangle</a>
%F A359019 For n <= 1, a(n)=1;
%F A359019 otherwise for odd n > 1, a(n)=(A002478(n) + A000930(n) + 2 * A002478((n - 1) / 2) + 2 * A002478((n - 3) / 2)) / 4
%F A359019 and for even n, a(n)=(A002478(n) + A000930(n) + 2 * A002478((n - 2) / 2) + 2 * A002478(n / 2)) / 4
%F A359019 Alternatively, from _Walter Trump_:
%F A359019 For n <= 1, a(n)=1;
%F A359019 otherwise for odd n > 1, a(n)=(A000930(2n) + A000930(n) + 2 * A000930(n - 1) + 2 * A000930(n - 3)) / 4
%F A359019 and for even n, a(n)=(A000930(2n) + 2 * A000930(n - 2) + 3 * A000930(n)) / 4
%e A359019 a(4) is 6 because of:
%e A359019   +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
%e A359019   | | | | |     | |   | | |   | | |   | | | | | |
%e A359019   +-+-+-+ +     + +   +-+ +   +-+ +   +-+ +-+-+-+
%e A359019   | | | | |     | |   | | |   | | |   | | |   | |
%e A359019   +-+-+-+ +     + +-+-+-+ +-+-+-+ +-+-+-+ +   +-+
%e A359019   | | | | |     | |   | | | |   | | | | | |   | |
%e A359019   +-+-+-+ +-+-+-+ +   +-+ +-+   + +-+-+-+ +-+-+-+
%e A359019   | | | | |     | |   | | | |   | | | | | | | | |
%e A359019   +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
%Y A359019 Column k = 3 of A227690.
%Y A359019 Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
%Y A359019 Fixed: A000045, A002478, A054856, A054857, A219925, A219926, A219927, A219928, A219929.
%Y A359019 Free: A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.
%Y A359019 Cf. A000930.
%K A359019 nonn
%O A359019 0,3
%A A359019 _John Mason_, Dec 12 2022