A363958 Expansion of (1 + x + x^3)/(1 - x^2 - 2*x^4 - 2*x^6 + x^8).

1, 1, 1, 2, 3, 4, 7, 10, 16, 23, 37, 53, 86, 123, 199, 285, 461, 660, 1068, 1529, 2474, 3542, 5731, 8205, 13276, 19007, 30754, 44030, 71242, 101996, 165033, 236275, 382301, 547334, 885605, 1267906, 2051515, 2937120

Offset: 0

Author

Greg Dresden and Yunxin Li, Jun 29 2023

Keywords

• nonn
• easy

Comments

a(n) is the number of ways to tile a zig-zag strip of n cells using squares (of length 1), strips (of length 3), and triangles (using 3 cells), where the zig-zag strip begins below the center line. Here is the zig-zag strip corresponding to n=12, with 12 cells:

_ _ _

| | | | | |

| _| |_ | _| |_

| | | | | | |

|__|___|_|___| |___|

| | | | | |

|_| |_| |_|,

and here are the three possible triangles and strips (which can also be rotated or reflected):

_

| |

| | _

| | | |

| _| _ ___ _ | |

| | | | | |

|_|, |_ ___ _|, |_ ___|.

As an example, here is one of the a(12) = 86 ways to tile the skew double-strip of 12 cells:

_ _ _

| | | | | |

| |___| |___|_|_

| | | |

| _|_ |__________|

| | | | | |

|_| |_| |_|.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1,0,2,0,2,0,1).

Crossrefs

Cf. A077998. Alternate terms are A123392 and A210460.

Programs

Mathematica

LinearRecurrence[{0, 1, 0, 2, 0, 2, 0, 1}, {1, 1, 1, 2, 3, 4, 7, 10}, 50]

Formula

a(n) = a(n-2) + 2*a(n-4) + 2*a(n-6) + a(n-8).

a(2*n) = a(2*n-1) + a(2*n-3) + a(2*n-5).

a(2*n) = A123392(n).

a(2*n+1) = a(2*n) + a(2*n-2).

a(2*n+1) = A210460(n+1).