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A329185 Number of ways to tile a 2 X n grid with dominoes and L-trominoes such that no four tiles meet at a corner.

%I A329185 #26 Apr 08 2024 18:52:57
%S A329185 1,1,2,5,10,22,49,105,227,494,1071,2322,5038,10927,23699,51405,111498,
%T A329185 241837,524546,1137742,2467761,5352577,11609747,25181550,54618807,
%U A329185 118468250,256957750,557341615,1208874523,2622050045,5687229162,12335605733,26755941146
%N A329185 Number of ways to tile a 2 X n grid with dominoes and L-trominoes such that no four tiles meet at a corner.
%C A329185 a(n) <= A052980(n).
%H A329185 Peter Kagey, <a href="/A329185/b329185.txt">Table of n, a(n) for n = 0..2500</a>
%H A329185 Misha Lavrov, <a href="https://math.stackexchange.com/a/3426264/121988">Number of ways to tile a room with I-Shaped and L-Shaped Tiles</a>, Mathematics Stack Exchange.
%H A329185 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,3,-1,2).
%F A329185 a(n) = 2*a(n-1) - a(n-2) + 3*a(n-3) - a(n-4) + 2*a(n-5), with a(0) = a(1) = 1, a(2) = 2, a(3) = 5, and a(4) = 10.
%F A329185 G.f.: (1 - x)*(1 + x^2) / (1 - 2*x + x^2 - 3*x^3 + x^4 - 2*x^5). - _Colin Barker_, Nov 12 2019
%e A329185 For n=3, the five tilings are:
%e A329185 +---+---+---+  +---+---+---+
%e A329185 |   |   |   |  |   |       |
%e A329185 +   +   +   +  +   +---+---+
%e A329185 |   |   |   |  |   |       |
%e A329185 +---+---+---+, +---+---+---+,
%e A329185 +---+---+---+  +---+---+---+
%e A329185 |       |   |  |   |       |
%e A329185 +---+---+   +  +   +---+   +
%e A329185 |       |   |  |       |   |
%e A329185 +---+---+---+, +---+---+---+, and
%e A329185 +---+---+---+
%e A329185 |       |   |
%e A329185 +   +---+   +
%e A329185 |   |       |
%e A329185 +---+---+---+.
%e A329185 For n=4, the only tiling counted by A052980(4) that is not counted by a(4) is
%e A329185 +---+---+---+---+
%e A329185 |       |       |
%e A329185 +---+---+---+---+
%e A329185 |       |       |
%e A329185 +---+---+---+---+.
%t A329185 LinearRecurrence[{2, -1, 3, -1, 2}, {1, 1, 2, 5, 10}, 50] (* _Paolo Xausa_, Apr 08 2024 *)
%o A329185 (PARI) Vec((1 - x)*(1 + x^2) / (1 - 2*x + x^2 - 3*x^3 + x^4 - 2*x^5) + O(x^30)) \\ _Colin Barker_, Nov 12 2019
%Y A329185 A052980 is the analogous problem without the "four corners" restriction.
%K A329185 nonn,easy
%O A329185 0,3
%A A329185 _Peter Kagey_, Nov 07 2019