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A348134 Number of ways two L-tiles (with rotation) can be placed on an n X n square.

%I A348134 #54 Aug 05 2025 12:10:54
%S A348134 0,0,22,336,1422,3952,8790,16992,29806,48672,75222,111280,158862,
%T A348134 220176,297622,393792,511470,653632,823446,1024272,1259662,1533360,
%U A348134 1849302,2211616,2624622,3092832,3620950,4213872,4876686,5614672,6433302,7338240,8335342,9430656
%N A348134 Number of ways two L-tiles (with rotation) can be placed on an n X n square.
%C A348134 All terms are even, because groups of ways, which are connected by 90 degrees rotation symmetry, are made up from 4 or 2 ways, so the number of ways will be some 4m+2n, and 4m+2n is even.
%H A348134 Nicolas Bělohoubek, <a href="/A348134/a348134.pdf">Visualization of 3rd term</a>
%H A348134 Nicolas Bělohoubek, <a href="/A348134/a348134_1.pdf">90° rotation groups for 3rd term</a>
%H A348134 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A348134 a(n) = 2*(n - 2)*(4*n^3 - 8*n^2 - 19*n + 32) for n > 1.
%F A348134 G.f.: 2*x^3*(11 + 113*x - 19*x^2 - 9*x^3)/(1 - x)^5. - _Stefano Spezia_, Oct 03 2021
%F A348134 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Wesley Ivan Hurt_, Aug 05 2025
%e A348134 For a(1) and a(2) there are fewer squares on the main square then squares of the 2 L-tiles, so a(1) = a(2) = 0.
%t A348134 LinearRecurrence[{5,-10,10,-5,1},{0,0,22,336,1422,3952},40] (* _Harvey P. Dale_, Mar 04 2023 *)
%Y A348134 Cf. A242856, A243645.
%K A348134 nonn,easy
%O A348134 1,3
%A A348134 _Nicolas Bělohoubek_, Oct 02 2021