cp's OEIS Frontend

A245596 Number of tilings of an n X 1 rectangle by tiles of dimension 1 X 1 and 2 X 1 such that every tile is next to a tile of different size.

%I A245596 #21 Jul 16 2020 05:29:24
%S A245596 1,0,0,2,1,1,4,3,4,8,9,12,19,24,33,48,64,88,124,169,233,324,445,614,
%T A245596 850,1171,1616,2233,3080,4251,5870,8100,11180,15434,21301,29401,40584,
%U A245596 56015,77316,106720,147301,203316,280635,387352,534653
%N A245596 Number of tilings of an n X 1 rectangle by tiles of dimension 1 X 1 and 2 X 1 such that every tile is next to a tile of different size.
%H A245596 Paul Tek, <a href="/A245596/b245596.txt">Table of n, a(n) for n = 0..1000</a>
%H A245596 Paul Tek, <a href="/A245596/a245596.jpg">Illustration of the formula</a>
%H A245596 Paul Tek, <a href="/A245596/a245596_1.png">Illustration of the first terms</a>
%H A245596 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1,1,1).
%F A245596                          [0 1 0 0 1 0 0]     [1]
%F A245596                          [0 0 1 0 0 0 0]     [0]
%F A245596                          [0 0 0 1 0 0 0]     [0]
%F A245596 a(n) = [1 0 0 0 0 0 0] * [0 0 0 0 1 0 1]^n * [1], for any n>=0.
%F A245596                          [0 0 0 0 0 1 0]     [0]
%F A245596                          [0 0 0 0 0 0 1]     [0]
%F A245596                          [0 1 1 0 0 0 0]     [1]
%F A245596 G.f.: (x^6+x^3+1)/(-x^6-x^5-x^4-x^3+1) = -(x^6+x^3+1)/((x^2+x+1)*(x^4+x-1)). - _Colin Barker_, Jul 27 2014
%F A245596 a(n) = a(n-3)+a(n-4)+a(n-5)+a(n-6) for n>6. - _Colin Barker_, Jul 28 2014
%e A245596 A 3 X 1 rectangle can be tiled in three ways:
%e A245596   +-+-+-+  +-+---+     +---+-+
%e A245596   | | | |, | |   | and |   | |.
%e A245596   +-+-+-+  +-+---+     +---+-+
%e A245596 The first tiling is not acceptable, as none of the 1 X 1 tiles is next to a 2 X 1 tile.
%e A245596 The second and third tilings are acceptable, as every 1 X 1 tile is next to a 2 X 1 tile and vice versa.
%e A245596 Hence, a(3)=2.
%o A245596 (PARI) Vec(-(x^6+x^3+1)/((x^2+x+1)*(x^4+x-1)) + O(x^100)) \\ _Colin Barker_, Jul 28 2014
%Y A245596 Cf. A230096.
%K A245596 nonn,easy
%O A245596 0,4
%A A245596 _Paul Tek_, Jul 27 2014