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Number of inequivalent tilings of a 4 X n rectangle using integer-sided square tiles.
11

%I #19 Mar 18 2023 11:35:13

%S 1,1,4,6,13,39,115,295,861,2403,7048,20377,60008,175978,519589,

%T 1532455,4531277,13395656,39639758,117301153,347248981,1028011708,

%U 3043852214,9012879842,26689014028,79033362580,234045889421,693101137571,2052569508948

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

%H John Mason, <a href="/A359020/b359020.txt">Table of n, a(n) for n = 0..1000</a>

%H John Mason, <a href="/A359019/a359019.pdf">Counting free tilings of a rectangle</a>

%F For even n > 4

%F a(n) = (A054856(n) + compo(n) + 4 * A054856((n - 2) / 2) +

%F 2 * A054856((n - 4) / 2) + 2 * A054856(n / 2) +

%F 2 * Sum_{k=0..(n - 2) / 2} (A054856(k))) / 4

%F For odd n > 4

%F a(n) = (A054856(n) + compo(n) + 2 * A054856((n - 3) / 2) +

%F 2 * A054856((n - 1) / 2) + 2 * Sum_ {k=0..(n - 3) / 2} (A054856(k))) / 4

%F Where compo(n) is the number of distinct compositions of n as a sum of 1, 2, (1+1) and 4.

%e a(3) is 6 because of:

%e +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+

%e | | | | | | | | | | | | | | | | | | |

%e +-+-+-+ + + + +-+ + +-+ + +-+ +-+-+-+

%e | | | | | | | | | | | | | | | | | |

%e +-+-+-+ + + +-+-+-+ +-+-+-+ +-+-+-+ + +-+

%e | | | | | | | | | | | | | | | | | | |

%e +-+-+-+ +-+-+-+ + +-+ +-+ + +-+-+-+ +-+-+-+

%e | | | | | | | | | | | | | | | | | | | |

%e +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+

%Y Column k = 4 of A227690.

%Y Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:

%Y Fixed: A000045, A002478, A054856, A054857, A219925, A219926, A219927, A219928, A219929.

%Y Free: A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.

%K nonn

%O 0,3

%A _John Mason_, Dec 12 2022