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A286442
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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.
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7
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0, 57, 9233, 563287, 12649059, 152516103, 1211235921, 7147857411, 33812251267, 134823778299, 469266000129, 1462057867743, 4154650828483, 10922915001087, 26867398129329, 62381437357035, 137705497065315, 290721776912275, 589883390417697, 1155073034088999, 2190429436721571
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OFFSET
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5,2
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COMMENTS
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Rotations and reflections of tilings are counted. Tiles of the same size are not distinguishable.
For an analogous problem concerning square tiles, see A194788.
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LINKS
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Heinrich Ludwig, Table of n, a(n) for n = 5..100
Heinrich Ludwig, Illustration of tiling a 6X6X6 area
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).
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FORMULA
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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.
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
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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Cf. A194788, A286436, A286437, A286438, A286439, A286440, A286441.
Sequence in context: A218353 A127455 A263669 * A219077 A091749 A218425
Adjacent sequences: A286439 A286440 A286441 * A286443 A286444 A286445
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KEYWORD
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nonn,easy
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AUTHOR
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Heinrich Ludwig, May 15 2017
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STATUS
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approved
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