A286436 -id:A286436 - cp's OEIS frontend

A286437 Number of ways to tile an n X n X n triangular area with two 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-8) of 1 X 1 X 1 tiles.

0, 9, 48, 153, 372, 765, 1404, 2373, 3768, 5697, 8280, 11649, 15948, 21333, 27972, 36045, 45744, 57273, 70848, 86697, 105060, 126189, 150348, 177813, 208872, 243825, 282984, 326673, 375228, 428997, 488340, 553629, 625248, 703593, 789072, 882105, 983124, 1092573

Offset: 3

Heinrich Ludwig, May 10 2017

Rotations and reflections of tilings are counted. If they are to be ignored, see A286444. Tiles of the same size are indistinguishable.

For an analogous problem concerning square tiles, see A061995.

			There are 9 ways of tiling a triangular area of side 4 with two tiles of side 2 and an appropriate number (= 8) of tiles of side 1. See example in links section.

Heinrich Ludwig, Table of n, a(n) for n = 3..100
Heinrich Ludwig, Illustration of tiling a 4X4X4 area
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A286436, A286444, A286438, A286439, A286440, A286441, A286442, A061995.

concat(0, Vec(3*x^4*(3 + x + x^2 - x^3) / (1 - x)^5 + O(x^60))) \\ Colin Barker, May 12 2017

a(n) = (n^4 - 6*n^3 + 5*n^2 + 30*n - 54)/2, n>=3.
From Colin Barker, May 12 2017: (Start)
G.f.: 3*x^4*(3 + x + x^2 - x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>7.
(End)

A286438 Number of 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.

Original entry on oeis.org

0, 4, 63, 494, 2247, 7396, 19739, 45518, 94259, 179732, 321031, 543774, 881423, 1376724, 2083267, 3067166, 4408859, 6205028, 8570639, 11641102, 15574551, 20554244, 26791083, 34526254, 44033987, 55624436, 69646679, 86491838, 106596319, 130445172, 158575571, 191580414

Offset: 3

Heinrich Ludwig, May 11 2017

Rotations and reflections of tilings are counted. If they are to be ignored, see A286445. Tiles of the same size are not distinguishable.

For an analogous problem concerning square tiles, see A061996.

			There are 4 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.

Heinrich Ludwig, Table of n, a(n) for n = 3..100
Heinrich Ludwig, Example for n=4
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A286436, A286445, A286437, A286439, A286440, A286441, A286442, A061996.

concat(0, Vec(x^4*(4 + 35*x + 137*x^2 - 28*x^3 - 24*x^4 - 15*x^5 + 11*x^6) / (1 - x)^7 + O(x^30))) \\ Colin Barker, May 11 2017

a(n) = (n^6 - 9*n^5 + 6*n^4 + 153*n^3 - 361*n^2 - 564*n + 1848)/6 for n>=4.

G.f.: x^4*(4 + 35*x + 137*x^2 - 28*x^3 - 24*x^4 - 15*x^5 + 11*x^6) / (1 - x)^7. - Colin Barker, May 11 2017

A286439 Number of 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.

Original entry on oeis.org

0, 1, 25, 747, 7459, 42983, 176373, 575775, 1595487, 3908979, 8701313, 17936083, 34713675, 63739327, 111921149, 189119943, 309074343, 490526475, 758575017, 1146284219, 1696579123, 2464458903, 3519561925, 4949117807, 6861323439, 9389181603, 12694842513, 16974490275

Offset: 3

Heinrich Ludwig, May 11 2017

Rotations and reflections of tilings are counted. If they are to be ignored, see A286446. Tiles of the same size are not distinguishable.

For an analogous problem concerning square tiles, see A061997.

			There are 25 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 example in links section.

Heinrich Ludwig, Table of n, a(n) for n = 3..100
Heinrich Ludwig, Illustration of tiling a 5X5X5 area
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A286436, A286446, A286437, A286438, A286440, A286441, A286442, A061997.

concat(0, Vec(x^4*(1 + 16*x + 558*x^2 + 1552*x^3 + 770*x^4 - 1674*x^5 + 306*x^6 + 144*x^7 + 45*x^8 - 38*x^9) / (1 - x)^9 + O(x^60))) \\ Colin Barker, May 12 2017

a(n) = (n^8 -12*n^7 +6*n^6 +432*n^5 -1279*n^4 -4692*n^3 +20592*n^2 +13320*n -91800)/24, for n>=5.

G.f.: x^4*(1 + 16*x + 558*x^2 + 1552*x^3 + 770*x^4 - 1674*x^5 + 306*x^6 + 144*x^7 + 45*x^8 - 38*x^9) / (1 - x)^9. - Colin Barker, May 12 2017

A286440 Number of ways to tile an n X n X n triangular area with five 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-20) of 1 X 1 X 1 tiles.

Original entry on oeis.org

0, 546, 14064, 157248, 1056516, 5086902, 19399860, 62311740, 175452816, 445146906, 1037833944, 2255992584, 4622997276, 9007684494, 16802136156, 30169344996, 52381036968, 88270019922, 144826036032, 231969248016, 363541216308, 558559556262, 842789431428, 1250692671180

Offset: 5

Heinrich Ludwig, May 12 2017

Rotations and reflections of tilings are counted. Tiles of the same size are not distinguishable.

For an analogous problem concerning square tiles, see A061998.

			There are 546 ways of tiling a triangular area of side 6 with 5 tiles of side 2 and an appropriate number (= 16) of tiles of side 1. See illustration in links section.

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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A061998, A286436, A286437, A286438, A286439, A286441, A286442.

concat(0, Vec(6*x^6*(91 + 1343*x + 5429*x^2 + 1703*x^3 - 4419*x^4 - 789*x^5 + 2379*x^6 - 627*x^7 - 76*x^8 - 14*x^9 + 20*x^10) / (1 - x)^11 + O(x^40))) \\ Colin Barker, May 12 2017

a(n) = (n^10 -15*n^9 +5*n^8 +930*n^7 -3325*n^6 -19863*n^5 +109915*n^4 +155100*n^3 -1365876*n^2 -191592*n +5981760)/120 for n >= 6.

G.f.: 6*x^6*(91 + 1343*x + 5429*x^2 + 1703*x^3 - 4419*x^4 - 789*x^5 + 2379*x^6 - 627*x^7 - 76*x^8 - 14*x^9 + 20*x^10) / (1 - x)^11. - Colin Barker, May 12 2017

A286441 Number of ways to tile an n X n X n triangular area with six 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-24) of 1 X 1 X 1 tiles.

Original entry on oeis.org

0, 219, 15160, 369787, 4366982, 32450843, 175628996, 755759531, 2734928266, 8643796747, 24503068784, 63522668395, 152816062222, 345005930315, 737473609532, 1503178571195, 2938515130514, 5535661080283, 10089397100584, 17851538034587, 30750030827926, 51694565135803

Offset: 5

Heinrich Ludwig, May 12 2017

Rotations and reflections of tilings are counted. Tiles of the same size are not distinguishable.

For an analogous problem concerning square tiles, see A172158.

			There are 219 ways of tiling a triangular area of side 6 with 6 tiles of side 2 and an appropriate number (= 12) of tiles of side 1. See illustration in links section.

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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A172158, A286436, A286437, A286438, A286439, A286440, A286442.

concat(0, Vec( x^6*(219 + 12313*x + 189789*x^2 + 679597*x^3 + 344288*x^4 - 808902*x^5 + 54074*x^6 + 289970*x^7 - 51453*x^8 - 71891*x^9 + 27785*x^10 - 255*x^11 + 98*x^12 - 352*x^13) / (1 - x)^13 + O(x^60))) \\ Colin Barker, May 13 2017

a(n) = (n^12 - 18*n^11 + 3*n^10 + 1710*n^9 - 7175*n^8 - 60078*n^7 + 401649*n^6 + 884466*n^5 - 9521846*n^4 - 3238224*n^3 + 107453448*n^2 - 25651296*n - 483140880)/720 for n >= 7.

G.f.: x^6*(219 + 12313*x + 189789*x^2 + 679597*x^3 + 344288*x^4 - 808902*x^5 + 54074*x^6 + 289970*x^7 - 51453*x^8 - 71891*x^9 + 27785*x^10 - 255*x^11 + 98*x^12 - 352*x^13) / (1 - x)^13. - Colin Barker, May 13 2017

A286442 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.

Original entry on oeis.org

0, 57, 9233, 563287, 12649059, 152516103, 1211235921, 7147857411, 33812251267, 134823778299, 469266000129, 1462057867743, 4154650828483, 10922915001087, 26867398129329, 62381437357035, 137705497065315, 290721776912275, 589883390417697, 1155073034088999, 2190429436721571

Offset: 5

Heinrich Ludwig, May 15 2017

Rotations and reflections of tilings are counted. Tiles of the same size are not distinguishable.

For an analogous problem concerning square tiles, see A194788.

			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.

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).

Cf. A194788, A286436, A286437, A286438, A286439, A286440, A286441.

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

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

A286443 Irregular triangle read by rows: T(n, k) = number of non-equivalent ways to tile an n X n X n triangular area with k 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-4*k) of 1 X 1 X 1 tiles.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 1, 4, 10, 14, 6, 1, 6, 32, 97, 142, 105, 46, 14, 3, 1, 1, 8, 70, 398, 1280, 2386, 2574, 1569, 524, 87, 3, 1, 11, 143, 1290, 7301, 26471, 62067, 94423, 93358, 60287, 25881, 7697, 1678, 281, 40, 5, 1, 1, 13, 252, 3366, 29603, 176591, 728868

Offset: 1

Heinrich Ludwig, May 16 2017

The triangle T(n, k) is irregularly shaped: For n >= 4: 0 <= k <= n^2/4 if n is even, 0 <= k <= (n^2 -9)/4 if n is odd. First row corresponds to n = 1.
Rotations and reflections of tilings are not counted. If they are to be counted, see A286436. Tiles of the same size are indistinguishable.
For an analogous problem concerning square tiles, see A236679.

			The triangle begins with T(1, 0)
   1;
   1,    1;
   1,    1;
   1,    3,    3,    2,    1;
   1,    4,   10,   14,    6;
   1,    6,   32,   97,  142,  105,   46,   14,    3,    1;
   1,    8,   70,  398, 1280, 2386, 2574, 1569,  524,   87,    3;
T(4, 3) = 2 because there are 2 non-equivalent ways to tile an area of size 4X4X4 with 3 tiles of size 2X2X2 and fill up the rest with tiles of size 1X1X1.

Heinrich Ludwig, Table of n, a(n) for n = 1..140

Cf. A236679, A286436, A286444, A286445, A286446.

cp's OEIS Frontend

A286437 Number of ways to tile an n X n X n triangular area with two 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-8) of 1 X 1 X 1 tiles.

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A286438 Number of 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.

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A286439 Number of 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.

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A286440 Number of ways to tile an n X n X n triangular area with five 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-20) of 1 X 1 X 1 tiles.

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A286441 Number of ways to tile an n X n X n triangular area with six 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-24) of 1 X 1 X 1 tiles.

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A286442 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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A286443 Irregular triangle read by rows: T(n, k) = number of non-equivalent ways to tile an n X n X n triangular area with k 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-4*k) of 1 X 1 X 1 tiles.

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