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%I #12 May 13 2017 03:46:06
%S 0,2,14,97,398,1290,3366,7731,15888,30248,53850,91147,147496,230290,
%T 348148,512457,736204,1035986,1430420,1942691,2598470,3429064,4468784,
%U 5758755,7343670,9276330,11613714,14422313,17773458,21749506,26438362,31940587,38363044,45826992
%N Number of non-equivalent ways to tile an n X n X n triangular area with three 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-12) of 1 X 1 X 1 tiles.
%C Rotations and reflections of tilings are not counted. If they are to be counted, see A286438. Tiles of the same size are indistinguishable.
%C For an analogous problem concerning square tiles, see A279112.
%H Heinrich Ludwig, <a href="/A286445/b286445.txt">Table of n, a(n) for n = 3..100</a>
%H Heinrich Ludwig, <a href="/A286445/a286445.png">Illustration of tiling a 4X4X4 area</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).
%F a(n) = (n^6 -9*n^5 +6*n^4 +165*n^3 -447*n^2 -372*n +1736)/36 + IF(MOD(n, 2) = 1, -n^2 +6*n -9)/2 + IF(MOD(n, 3) = 0, -2)/9 for n >= 4.
%F G.f.: x^4*(2 + 8*x + 55*x^2 + 121*x^3 + 188*x^4 + 121*x^5 + 44*x^6 - 39*x^7 - 22*x^8 - 5*x^9 + 5*x^10 + 2*x^11) / ((1 - x)^7*(1 + x)^3*(1 + x + x^2)). - _Colin Barker_, May 12 2017
%e There are 2 non-equivalent ways of tiling a triangular area of side 4 with three tiles of side 2 and an appropriate number (= 4) of tiles of side 1. See example in links section.
%o (PARI) concat(0, Vec(x^4*(2 + 8*x + 55*x^2 + 121*x^3 + 188*x^4 + 121*x^5 + 44*x^6 - 39*x^7 - 22*x^8 - 5*x^9 + 5*x^10 + 2*x^11) / ((1 - x)^7*(1 + x)^3*(1 + x + x^2)) + O(x^60))) \\ _Colin Barker_, May 12 2017
%Y Cf. A279112, A286438, A286443, A286444, A286446.
%K nonn,easy
%O 3,2
%A _Heinrich Ludwig_, May 12 2017
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