%I #15 Jun 13 2015 00:55:16

%S 1,0,1,1,2,1,4,3,7,7,13,14,24,28,45,56,86,111,165,218,317,426,611,831,

%T 1181,1619,2286,3150,4428,6123,8582,11896,16641,23105,32278,44865,

%U 62620,87103,121499,169087,235761,328214,457508,637064,887857,1236500,1723054

%N Number of tilings of an n X 1 rectangle by tiles of dimension 1 X 1 and 2 X 1 such that every tile shares an equal-length edge with a tile of the same size.

%H Paul Tek, <a href="/A248880/b248880.txt">Table of n, a(n) for n = 0..1000</a>

%H Paul Tek, <a href="/A248880/a248880.png">Illustration of the formula</a>

%H Paul Tek, <a href="/A248880/a248880_1.png">Illustration of the first terms</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,0,1).

%F [ 0 1 0 1 0 0 0 ] [1]

%F [ 0 0 1 0 0 0 0 ] [0]

%F [ 0 0 1 1 0 0 0 ] [1]

%F a(n) = [1 0 0 0 0 0 0] * [ 0 0 0 0 1 0 0 ] ^ n * [0], for any n>=0.

%F [ 0 0 0 0 0 1 0 ] [0]

%F [ 0 0 0 0 0 0 1 ] [0]

%F [ 0 1 0 0 0 1 0 ] [1]

%F G.f.: -(x^2-x+1)*(x^4-x^2+1) / (x^6-x^3+x^2+x-1). - _Colin Barker_, Mar 05 2015

%e A 3 X 1 rectangle can be tiled in three ways:

%e +-+-+-+ +-+---+ +---+-+

%e | | | |, | | | and | | |.

%e +-+-+-+ +-+---+ +---+-+

%e The first tiling is acceptable, as every 1 X 1 tile is next to another 1 X 1 tile (and there are no 2 X 1 tiles).

%e The second and third tilings are not acceptable, as the 1 X 1 tiles are not next to other 1 X 1 tiles.

%e Hence, a(3)=1.

%o (PARI) Vec(-(x^2-x+1)*(x^4-x^2+1)/(x^6-x^3+x^2+x-1) + O(x^100)) \\ _Colin Barker_, Mar 05 2015

%Y Cf. A245596.

%K nonn,easy

%O 0,5

%A _Paul Tek_, Mar 05 2015