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A248880 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.

%I A248880 #15 Jun 13 2015 00:55:16
%S A248880 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 A248880 1181,1619,2286,3150,4428,6123,8582,11896,16641,23105,32278,44865,
%U A248880 62620,87103,121499,169087,235761,328214,457508,637064,887857,1236500,1723054
%N A248880 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 A248880 Paul Tek, <a href="/A248880/b248880.txt">Table of n, a(n) for n = 0..1000</a>
%H A248880 Paul Tek, <a href="/A248880/a248880.png">Illustration of the formula</a>
%H A248880 Paul Tek, <a href="/A248880/a248880_1.png">Illustration of the first terms</a>
%H A248880 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,0,1).
%F A248880                          [ 0 1 0 1 0 0 0 ]       [1]
%F A248880                          [ 0 0 1 0 0 0 0 ]       [0]
%F A248880                          [ 0 0 1 1 0 0 0 ]       [1]
%F A248880 a(n) = [1 0 0 0 0 0 0] * [ 0 0 0 0 1 0 0 ] ^ n * [0], for any n>=0.
%F A248880                          [ 0 0 0 0 0 1 0 ]       [0]
%F A248880                          [ 0 0 0 0 0 0 1 ]       [0]
%F A248880                          [ 0 1 0 0 0 1 0 ]       [1]
%F A248880 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 A248880 A 3 X 1 rectangle can be tiled in three ways:
%e A248880   +-+-+-+  +-+---+     +---+-+
%e A248880   | | | |, | |   | and |   | |.
%e A248880   +-+-+-+  +-+---+     +---+-+
%e A248880 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 A248880 The second and third tilings are not acceptable, as the 1 X 1 tiles are not next to other 1 X 1 tiles.
%e A248880 Hence, a(3)=1.
%o A248880 (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 A248880 Cf. A245596.
%K A248880 nonn,easy
%O A248880 0,5
%A A248880 _Paul Tek_, Mar 05 2015