A366143 a(n) = a(n-2) + 2*a(n-4) - a(n-10), with a[0..9] = [1, 1, 1, 1, 1, 2, 3, 5, 6, 9].

1, 1, 1, 1, 1, 2, 3, 5, 6, 9, 11, 18, 22, 35, 43, 69, 84, 134, 164, 263, 321, 513, 627, 1004, 1226, 1961, 2396, 3835, 4684, 7494, 9155, 14651, 17896, 28635, 34980, 55976, 68376, 109411, 133652, 213869, 261249, 418040, 510657, 817143, 998175, 1597247, 1951113

Offset: 0

Author

Greg Dresden and Ziyi Xie, Sep 30 2023

Keywords

• nonn
• easy

Comments

a(n) is the number of ways to tile a zig-zag strip of n cells using squares (of 1 cell) and strips (of 3 cells). Here is the zig-zag strip corresponding to n=11, with 11 cells:

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and here is the strip of 3 cells (which can be reflected)

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As an example, here is one of the a(11) = 18 ways to tile the zig-zag strip of 11 cells:

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Links

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

Crossrefs

Cf. A135318.

Programs

Mathematica

LinearRecurrence[{0, 1, 0, 2, 0, 0, 0, 0, 0, -1}, {1, 1, 1, 1, 1, 2,
  3, 5, 6, 9}, 40]

Formula

a(n) = a(n-2) + 2*a(n-4) - a(n-10).

a(2*n) = a(2*n-1) + a(2*n-4) - a(2*n-5) + a(2*n-6).

a(2*n+1) = a(2*n) + 2*a(2*n-3) - a(2*n-4) + a(2*n-6) - a(2*n-7).

G.f.: (x^8+x^7-x^5-2*x^4+x+1)/(x^10-2*x^4-x^2+1).