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A386489 Expansion of (1-x)/((1+x+2*x^2)*(1-4*x+x^2)).

%I A386489 #31 Aug 26 2025 07:13:45
%S A386489 1,2,7,30,109,402,1511,5638,21021,78474,292887,1093006,4079181,
%T A386489 15223810,56815879,212039702,791343293,2953333114,11021988791,
%U A386489 41134623134,153516503405,572931388658,2138209053735,7979904827430,29781410249821,111145736175722
%N A386489 Expansion of (1-x)/((1+x+2*x^2)*(1-4*x+x^2)).
%C A386489 a(n) is the number of ways to tile a 2 X n board with squares, dominoes, and L-shaped quadrominoes. Here is one of the a(4)=109 possible tilings of a 2 X 4 board:
%C A386489     _______
%C A386489    | | |_|_|
%C A386489    |_|_____|
%C A386489 Compare to A030186 which counts the tilings with just squares and dominos.
%H A386489 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,7,-2).
%F A386489 a(n) = 3*a(n-1) + a(n-2) + 7*a(n-3) - 2*a(n-4).
%F A386489 a(n) = A030186(n) + 2*sum_{i=0..n-2}(A033505(n-i-3)*a(i) + A030186(n-i-3)*(a(i)+2*sum_{j=0..i} a(j)).
%F A386489 a(n) ~ (2 + sqrt(3))^(n+2) / (18 + 4*sqrt(3)). - _Vaclav Kotesovec_, Aug 21 2025
%F A386489 23*a(n) = -4*A001353(n)+13*A001353(n+1) +10*A001607(n+1)+8*A001607(n) . - _R. J. Mathar_, Aug 26 2025
%t A386489 LinearRecurrence[{3, 1, 7, -2}, {1, 2, 7, 30}, 30]
%Y A386489 Cf. A030186, A033505.
%K A386489 nonn,easy,new
%O A386489 0,2
%A A386489 _Greg Dresden_ and Madison Lingchen Zhou, Aug 20 2025