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A334396 Number of fault-free tilings of a 3 X n rectangle with squares and dominoes.

%I A334396 #54 Jun 15 2022 01:51:10
%S A334396 0,0,2,2,10,16,52,104,286,634,1622,3768,9336,22152,54106,129610,
%T A334396 314546,756728,1831196,4413952,10667462,25735346,62160046,150020016,
%U A334396 362257392,874442064,2111291570,5096782418,12305249242,29706645280,71719568260
%N A334396 Number of fault-free tilings of a 3 X n rectangle with squares and dominoes.
%C A334396 A fault-free tiling has no horizontal or vertical faults (that is to say, the tiling does not split along any interior horizontal or vertical line).
%H A334396 Colin Barker, <a href="/A334396/b334396.txt">Table of n, a(n) for n = 1..1000</a>
%H A334396 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-1,-1).
%F A334396 a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4) for n >= 5.
%F A334396 a(n) = 2*A112577(n-2) for n >= 2.
%F A334396 G.f.: 2*x^3 / ((1 + x - x^2)*(1 - 2*x - x^2)). - _Colin Barker_, Aug 06 2020
%e A334396 a(4) = 2 because these are the only fault-free tilings of the 3 X 4 rectangle with squares and dominoes:
%e A334396 ._ _ _ _     _ _ _ _
%e A334396 |_ _|_| |   | |_|_ _|
%e A334396 | |_ _|_|   |_|_ _| |
%e A334396 |_|_|_ _|   |_ _|_|_|
%t A334396 a[n_]:= (2/3)*(Fibonacci[n-1, 2] - (-1)^n*Fibonacci[n-1]);
%t A334396 Table[a[n], {n, 40}] (* _G. C. Greubel_, Jan 15 2022 *)
%o A334396 (PARI) concat([0,0] , Vec(2*x^3/((1+x-x^2)*(1-2*x-x^2)) + O(x^30))) \\ _Colin Barker_, Aug 06 2020
%o A334396 (Magma) [n le 4 select 2*Floor((n-1)/2) else Self(n-1) +4*Self(n-2) -Self(n-3) -Self(n-4): n in [1..40]]; // _G. C. Greubel_, Jan 15 2022
%o A334396 (Sage) [(2/3)*(lucas_number1(n-1,2,-1) - (-1)^n*lucas_number1(n-1,1,-1)) for n in (1..40)] # _G. C. Greubel_, Jan 15 2022
%Y A334396 Cf. A084477, A084479, A084481, A112577, A124997, A335747.
%K A334396 nonn,easy
%O A334396 1,3
%A A334396 _Greg Dresden_ and _Oluwatobi Jemima Alabi_, Jul 06 2020