A334047 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.

0, 0, 6, 16, 35, 78, 168, 352, 735, 1535, 3201, 6670, 13897, 28952, 60311, 125632, 261698, 545127, 1135516, 2365311, 4927005, 10263077, 21378247, 44531422, 92760060, 193221509, 402485199, 838386656

Offset

1,3

Comments

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:

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Links

Table of n, a(n) for n=1..28.

Formula

Conjectures from Colin Barker, Sep 06 2020: (Start)

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)).

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.

(End)

Example

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.

Mathematica

B[1] = 0; B[2] = 0; B[3] = 0; B[4] = 0; B[5] = 5;
B[n_?IntegerQ] :=
  B[n] = Floor[(n - 3)/2]*n +
    Sum[Floor[(i + 1)/2]*B[n - 4 - i], {i, 1, n - 5}];
Table[B[2 n], {n, 1, 30}]

Crossrefs

Sequence in context: A083046 A192749 A160997 * A199629 A098943 A321973

Adjacent sequences: A334044 A334045 A334046 * A334048 A334049 A334050

Keyword

nonn

Author

Tianshu Ouyang and Greg Dresden, Sep 05 2020

Status

approved