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%I #7 May 17 2017 21:33:24
%S 0,57,9233,563287,12649059,152516103,1211235921,7147857411,
%T 33812251267,134823778299,469266000129,1462057867743,4154650828483,
%U 10922915001087,26867398129329,62381437357035,137705497065315,290721776912275,589883390417697,1155073034088999,2190429436721571
%N Number of ways to tile an n X n X n triangular area with seven 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-28) of 1 X 1 X 1 tiles.
%C Rotations and reflections of tilings are counted. Tiles of the same size are not distinguishable.
%C For an analogous problem concerning square tiles, see A194788.
%H Heinrich Ludwig, <a href="/A286442/b286442.txt">Table of n, a(n) for n = 5..100</a>
%H Heinrich Ludwig, <a href="/A286442/a286442.png">Illustration of tiling a 6X6X6 area</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).
%F a(n) = (n^14 -21*n^13 +2835*n^11 -13664*n^10 -147903*n^9 +1159368*n^8 +3480705*n^7 -44292941*n^6 -24613344*n^5 +908186412*n^4 -372748320*n^3 -9895978296*n^2 +5596762608*n +46620962640)/5040 for n>=8.
%F G.f.: x^6*(57 + 8378*x + 430777*x^2 + 5143284*x^3 + 17802143*x^4 + 7781860*x^5 - 20367093*x^6 - 406014*x^7 + 12253687*x^8 - 5320950*x^9 - 731329*x^10 + 627984*x^11 + 198177*x^12 - 135016*x^13 + 10557*x^14 - 198*x^15 + 976*x^16) / (1 - x)^15. - _Colin Barker_, May 16 2017
%e There are 57 ways of tiling a triangular area of side 6 with 7 tiles of side 2 and an appropriate number (= 8) of tiles of side 1. See illustration in links section.
%o (PARI) concat(0, Vec(x^6*(57 + 8378*x + 430777*x^2 + 5143284*x^3 + 17802143*x^4 + 7781860*x^5 - 20367093*x^6 - 406014*x^7 + 12253687*x^8 - 5320950*x^9 - 731329*x^10 + 627984*x^11 + 198177*x^12 - 135016*x^13 + 10557*x^14 - 198*x^15 + 976*x^16) / (1 - x)^15 + O(x^30))) \\ _Colin Barker_, May 16 2017
%Y Cf. A194788, A286436, A286437, A286438, A286439, A286440, A286441.
%K nonn,easy
%O 5,2
%A _Heinrich Ludwig_, May 15 2017
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