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%I #14 May 15 2017 10:47:05
%S 0,1,6,142,1280,7301,29603,96485,266636,652908,1452054,2992513,
%T 5789499,10629381,18660890,31530854,51525116,81772345,126449707,
%U 191075297,282794784,410784700,586640186,824912741,1143620051,1564946921,2115898646,2829194838,3744093216,4907506597
%N Number of non-equivalent ways to tile an n X n X n triangular area with four 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-16) of 1 X 1 X 1 tiles.
%C Rotations and reflections of tilings are not counted. If they are to be counted, see A286439. Tiles of the same size are indistinguishable.
%C For an analogous problem concerning square tiles, see A279113.
%H Heinrich Ludwig, <a href="/A286446/b286446.txt">Table of n, a(n) for n = 3..100</a>
%H Heinrich Ludwig, <a href="/A286446/a286446.png">Illustration of tiling a 5X5X5 area</a>
%H <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1).
%F a(n) = (n^8 -12*n^7 +6*n^6 +432*n^5 -1249*n^4 -5028*n^3 +21820*n^2 +12384*n -94000)/144 + IF(MOD(n, 2) = 1, -8*n^3 +72*n^2 -208*n +189)/24 + IF(MOD(n, 3) = 0, -n^2 +3*n +7)/9 for n >= 5.
%F G.f.: x^4*(1 + 4*x + 127*x^2 + 983*x^3 + 4353*x^4 + 11916*x^5 + 22875*x^6 + 31058*x^7 + 30066*x^8 + 18947*x^9 + 5576*x^10 - 2441*x^11 - 3003*x^12 - 698*x^13 + 707*x^14 + 536*x^15 + 71*x^16 - 73*x^17 - 37*x^18 - 8*x^19) / ((1 - x)^9*(1 + x)^4*(1 + x + x^2)^3). - _Colin Barker_, May 12 2017
%e There are 6 non-equivalent ways of tiling a triangular area of side 5 with 4 tiles of side 2 and an appropriate number (= 9) of tiles of side 1. See illustration in links section.
%o (PARI) concat(0, Vec(x^4*(1 + 4*x + 127*x^2 + 983*x^3 + 4353*x^4 + 11916*x^5 + 22875*x^6 + 31058*x^7 + 30066*x^8 + 18947*x^9 + 5576*x^10 - 2441*x^11 - 3003*x^12 - 698*x^13 + 707*x^14 + 536*x^15 + 71*x^16 - 73*x^17 - 37*x^18 - 8*x^19) / ((1 - x)^9*(1 + x)^4*(1 + x + x^2)^3) + O(x^60))) \\ _Colin Barker_, May 12 2017
%Y Cf. A279113, A286439, A286443, A286444, A286445.
%K nonn,easy
%O 3,3
%A _Heinrich Ludwig_, May 12 2017
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