A334047 - OEIS

%I #17 Jun 15 2022 01:51:33

%S 0,0,6,16,35,78,168,352,735,1535,3201,6670,13897,28952,60311,125632,

%T 261698,545127,1135516,2365311,4927005,10263077,21378247,44531422,

%U 92760060,193221509,402485199,838386656

%N a(n) is the number of tilings of a bracelet of length 2n with 1 color of 5-minoes and 6-minoes, 2 colors of 7-minoes and 8-minoes, 3 colors of 9-minoes and 10-minoes, and so on.

%C a(n) is also the number of ways to tile an unbreakable 3 X 2n bracelet with dominoes and with upside-down T-shaped tetrominoes which looks like this:

%C       _

%C     _| |_

%C    |_____|

%F Conjectures from _Colin Barker_, Sep 06 2020: (Start)

%F G.f.: x^3*(6 - 8*x + 7*x^2 - 2*x^3) / ((1 - x)*(1 - 3*x + 3*x^2 - 3*x^3 + 2*x^4 - x^5)).

%F a(n) = 4*a(n-1) - 6*a(n-2) + 6*a(n-3) - 5*a(n-4) + 3*a(n-5) - a(n-6) for n>6.

%F (End)

%e For n=5 the a(5) = 35 tilings are as follows: we can use 3 colors of 10-minoes, each of which can be rotated to 10 different positions or "phases", giving us 30, and we can use two (single-color) 5-minoes in five different "phases", giving us another 5 tilings, with a grand total of 30 + 5 = 35.

%t B[1] = 0; B[2] = 0; B[3] = 0; B[4] = 0; B[5] = 5;

%t B[n_?IntegerQ] :=

%t   B[n] = Floor[(n - 3)/2]*n +

%t     Sum[Floor[(i + 1)/2]*B[n - 4 - i], {i, 1, n - 5}];

%t Table[B[2 n], {n, 1, 30}]

%K nonn

%O 1,3

%A _Tianshu Ouyang_ and _Greg Dresden_, Sep 05 2020