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

0,6

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:

___ ___

___| |___| |___

| |___| |___| |___

|___| |___| |___| |

| |___| |___| |___|

|___| |___| |___|,

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

___

___| |

___| ___|

| ___|

|___|

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

___ ___

___| |___| |___

| |___| |___ |___

|___| ___| |___ |

| ___| |___| |___|

|___| |___| |___|

Links

Table of n, a(n) for n=0..46.

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

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).

Mathematica

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

Crossrefs

Cf. A135318.

Sequence in context: A241819 A227070 A032718 * A086191 A074658 A186106

Adjacent sequences: A366140 A366141 A366142 * A366144 A366145 A366146

Keyword

nonn,easy

Author

Greg Dresden and Ziyi Xie, Sep 30 2023

Status

approved