login
Number of inequivalent tilings of a 3 X n rectangle using integer-sided square tiles.
11

%I #22 Mar 18 2023 11:34:58

%S 1,1,2,3,6,10,21,39,82,163,347,717,1533,3232,6927,14748,31645,67690,

%T 145322,311535,668997,1435645,3083301,6619842,14218066,30533005,

%U 65580338,140847132,302522253,649759735,1395611508,2997573501,6438470626,13829057884,29703388721,63799607283,137035047576,294336860797,632205714741

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

%H John Mason, <a href="/A359019/b359019.txt">Table of n, a(n) for n = 0..1000</a>

%H John Mason, <a href="/A359019/a359019_1.pdf">Counting free tilings of a rectangle</a>

%F For n <= 1, a(n)=1;

%F otherwise for odd n > 1, a(n)=(A002478(n) + A000930(n) + 2 * A002478((n - 1) / 2) + 2 * A002478((n - 3) / 2)) / 4

%F and for even n, a(n)=(A002478(n) + A000930(n) + 2 * A002478((n - 2) / 2) + 2 * A002478(n / 2)) / 4

%F Alternatively, from _Walter Trump_:

%F For n <= 1, a(n)=1;

%F otherwise for odd n > 1, a(n)=(A000930(2n) + A000930(n) + 2 * A000930(n - 1) + 2 * A000930(n - 3)) / 4

%F and for even n, a(n)=(A000930(2n) + 2 * A000930(n - 2) + 3 * A000930(n)) / 4

%e a(4) is 6 because of:

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

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

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

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

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

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

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

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

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

%Y Column k = 3 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.

%Y Cf. A000930.

%K nonn

%O 0,3

%A _John Mason_, Dec 12 2022