cp's OEIS Frontend

A230096 Number of tilings of an n X 1 rectangle (using tiles of dimension 1 X 1 and 2 X 1) that share no tile at the same position with their mirrored image.

%I A230096 #55 Sep 15 2024 15:21:00
%S A230096 1,0,0,2,2,2,2,6,6,10,10,22,22,42,42,86,86,170,170,342,342,682,682,
%T A230096 1366,1366,2730,2730,5462,5462,10922,10922,21846,21846,43690,43690,
%U A230096 87382,87382,174762,174762,349526,349526,699050,699050,1398102,1398102,2796202
%N A230096 Number of tilings of an n X 1 rectangle (using tiles of dimension 1 X 1 and 2 X 1) that share no tile at the same position with their mirrored image.
%C A230096 For any k>0, it is possible to transform a pair of symmetric tilings of length 2*k-1 that share no tile with their mirrored image into a pair of symmetric tilings of length 2*k with the same property by inserting a 1 X 1 tile next to the central 2 X 1 tile :
%C A230096    +- ... -+---+-  ...  -+        +- ... -+---+-+-  ...  -+
%C A230096    |  ABC  |   |   XYZ   |        |  ABC  |   |X|   XYZ   |
%C A230096    +-  ..  +-+o+-+  ..  -+  <-->  +-  ..  +-+-o-+-+  ..  -+
%C A230096    |   ZYX   |   |  CBA  |        |   ZYX   |X|   |  CBA  |
%C A230096    +-  ...  -+---+- ... -+        +-  ...  -+-+---+- ... -+
%C A230096 This transformation is reversible, hence a(2*k-1) = a(2*k) for any k>0. - _Paul Tek_, Oct 15 2013
%H A230096 Paul Tek, <a href="/A230096/b230096.txt">Table of n, a(n) for n = 0..6646</a>
%H A230096 Paul Tek, <a href="/A230096/a230096.png">Illustration of the first terms</a>
%H A230096 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,2).
%F A230096                    [0 1 1]     [1]
%F A230096 a(2*k) = [1 0 0] * [1 0 1]^k * [0], for any k>=0.
%F A230096                    [1 1 0]     [0]
%F A230096                      [0 1 1]     [0]
%F A230096 a(2*k-1) = [1 0 0] * [1 0 1]^k * [1], for any k>=1.
%F A230096                      [1 1 0]     [1]
%F A230096 a(n) = a(n-2)+2*a(n-4). G.f.: -(2*x^3-x^2+1) / ((x^2+1)*(2*x^2-1)). - _Colin Barker_, Oct 14 2013
%F A230096 a(n) = A078008(floor((n+1)/2)). - _Ralf Stephan_, Oct 18 2013
%e A230096 A 5 x 1 rectangle can be tiled in 8 ways:
%e A230096   +-+-+-+-+-+
%e A230096 - |=|=|=|=|=| that shares 5 tiles with its mirrored image,
%e A230096   +-+-+-+-+-+
%e A230096   +-+-+-+---+
%e A230096 - | | |=|   | that shares 1 tile with its mirrored image,
%e A230096   +-+-+-+---+
%e A230096   +-+-+---+-+
%e A230096 - |=| |   |=| that shares 2 tiles with its mirrored image,
%e A230096   +-+-+---+-+
%e A230096   +-+---+-+-+
%e A230096 - |=|   | |=| that shares 2 tiles with its mirrored image,
%e A230096   +-+---+-+-+
%e A230096   +-+---+---+
%e A230096 - | |   |   | that shares no tile with its mirrored image,
%e A230096   +-+---+---+
%e A230096   +---+-+-+-+
%e A230096 - |   |=| | | that shares 1 tile with its mirrored image,
%e A230096   +---+-+-+-+
%e A230096   +---+-+---+
%e A230096 - | = |=| = | that shares 3 tiles with its mirrored image,
%e A230096   +---+-+---+
%e A230096   +---+---+-+
%e A230096 - |   |   | | that shares no tile with its mirrored image.
%e A230096   +---+---+-+
%e A230096 Hence, a(5)=2.
%o A230096 (PARI) M=[0,1,1;1,0,1;1,1,0];
%o A230096 a(n)=if(n%2==0, [1,0,0]*M^(n/2)*[1;0;0], [1,0,0]*M^((n-1)/2)*[0;1;1])[1]
%o A230096 (PARI) Vec(-(2*x^3-x^2+1)/((x^2+1)*(2*x^2-1)) + O(x^100)) \\ _Colin Barker_, Oct 15 2013
%Y A230096 Cf. A224918, A225202.
%K A230096 nonn,easy
%O A230096 0,4
%A A230096 _Paul Tek_, Oct 13 2013