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A335274 a(n) = 2*a(n-1) + a(n-3), where a(0) = 0, a(1) = 1, a(2) = 4.

%I A335274 #23 Mar 20 2025 06:26:47
%S A335274 0,1,4,8,17,38,84,185,408,900,1985,4378,9656,21297,46972,103600,
%T A335274 228497,503966,1111532,2451561,5407088,11925708,26302977,58013042,
%U A335274 127951792,282206561,622426164,1372804120,3027814801,6678055766,14728915652,32485646105,71649347976
%N A335274 a(n) = 2*a(n-1) + a(n-3), where a(0) = 0, a(1) = 1, a(2) = 4.
%C A335274 a(n) is the number of ways to tile a 2 x n strip, with a bent tromino added to the top, with dominos and L-shaped trominos:
%C A335274    _
%C A335274   |_|_
%C A335274   |_|_|_ _ _
%C A335274   |_|_|_|_|_| . . .
%C A335274   |_|_|_|_|_| . . .
%H A335274 Colin Barker, <a href="/A335274/b335274.txt">Table of n, a(n) for n = 0..1000</a>
%H A335274 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1).
%F A335274 a(n) = 2*a(n-1) + a(n-3).
%F A335274 a(n) = 2*A008998(n-1) - A008998(n-4).
%F A335274 a(n) = A008998(n-1) + 2*A008998(n-2).
%F A335274 G.f.: x*(1 + 2*x) / (1 - 2*x - x^3). - _Colin Barker_, Jun 04 2020
%e A335274 a(2) = 4 as shown by these four tilings:
%e A335274    _         _         _         _
%e A335274   |X|_      | |_      |X|_      | |_
%e A335274   |X|X|  ,  |_|X|  ,  |X|X|  ,  |_| |
%e A335274   |_ _|     |X X|     | | |     |X|_|
%e A335274   |_ _|     |_ _|     |_|_|     |X X|
%t A335274 LinearRecurrence[{2, 0, 1}, {0, 1, 4}, 50] (* _Paolo Xausa_, Mar 20 2025 *)
%o A335274 (PARI) concat(0, Vec(x*(1 + 2*x) / (1 - 2*x - x^3) + O(x^35))) \\ _Colin Barker_, Jun 04 2020
%Y A335274 Cf. A008998, A335242.
%K A335274 nonn,easy
%O A335274 0,3
%A A335274 _Michael Tulskikh_, May 30 2020