A286443 -id:A286443 - cp's OEIS frontend

A286436 Irregular triangle read by rows: T(n, k) = number of 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.

1, 1, 1, 1, 3, 1, 7, 9, 4, 1, 1, 13, 48, 63, 25, 1, 21, 153, 494, 747, 546, 219, 57, 9, 1, 1, 31, 372, 2247, 7459, 14064, 15160, 9233, 3069, 480, 14, 1, 43, 765, 7396, 42983, 157248, 369787, 563287, 556932, 358974, 153520, 45282, 9634, 1529, 186, 16, 1, 1, 57, 1404

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 counted. If they are to be ignored, see A286443. Tiles of the same size are indistinguishable.
For an analogous problem concerning square tiles, see A193580.

			The triangle begins with T(1, 0):
1;
1,  1;
1,  3;
1,  7,   9,   4,   1;
1, 13,  48,  63,  25;
1, 21, 153, 494, 747, 546, 219, 57, 9, 1;
T(4, 3) = 4 because there are 4 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. A193580, A286443, A286437, A286438, A286439, A286440, A286441, A286442.

A286444 Number of non-equivalent 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.

Original entry on oeis.org

0, 3, 10, 32, 70, 143, 252, 424, 660, 995, 1430, 2008, 2730, 3647, 4760, 6128, 7752, 9699, 11970, 14640, 17710, 21263, 25300, 29912, 35100, 40963, 47502, 54824, 62930, 71935, 81840, 92768, 104720, 117827, 132090, 147648, 164502, 182799, 202540, 223880, 246820, 271523

Offset: 3

Heinrich Ludwig, May 12 2017

Rotations and reflections of tilings are not counted. If they are to be counted, see A286437. Tiles of the same size are indistinguishable.

For an analogous problem concerning square tiles, see A279111.

			There are 3 non-equivalent 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 (3,-1,-5,5,1,-3,1).

Cf. A279111, A286437, A286443, A286445, A286446.

concat(0, Vec(x^4*(3 + x + 5*x^2 - x^3) / ((1 - x)^5*(1 + x)^2) + O(x^30))) \\ Colin Barker, May 12 2017

a(n) = (n^4 -6*n^3 +11*n^2 -12)/12 + IF(MOD(n, 2) = 1, -n +2)/2.

G.f.: x^4*(3 + x + 5*x^2 - x^3) / ((1 - x)^5*(1 + x)^2). - Colin Barker, May 12 2017

A286445 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.

Original entry on oeis.org

0, 2, 14, 97, 398, 1290, 3366, 7731, 15888, 30248, 53850, 91147, 147496, 230290, 348148, 512457, 736204, 1035986, 1430420, 1942691, 2598470, 3429064, 4468784, 5758755, 7343670, 9276330, 11613714, 14422313, 17773458, 21749506, 26438362, 31940587, 38363044, 45826992

Offset: 3

Heinrich Ludwig, May 12 2017

Rotations and reflections of tilings are not counted. If they are to be counted, see A286438. Tiles of the same size are indistinguishable.

For an analogous problem concerning square tiles, see A279112.

			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.

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 (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

Cf. A279112, A286438, A286443, A286444, A286446.

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

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.

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

A286446 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.

Original entry on oeis.org

0, 1, 6, 142, 1280, 7301, 29603, 96485, 266636, 652908, 1452054, 2992513, 5789499, 10629381, 18660890, 31530854, 51525116, 81772345, 126449707, 191075297, 282794784, 410784700, 586640186, 824912741, 1143620051, 1564946921, 2115898646, 2829194838, 3744093216, 4907506597

Offset: 3

Heinrich Ludwig, May 12 2017

Rotations and reflections of tilings are not counted. If they are to be counted, see A286439. Tiles of the same size are indistinguishable.

For an analogous problem concerning square tiles, see A279113.

			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.

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 (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1).

Cf. A279113, A286439, A286443, A286444, A286445.

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

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.

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

cp's OEIS Frontend

A286436 Irregular triangle read by rows: T(n, k) = number of 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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A286444 Number of non-equivalent 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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A286445 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.

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A286446 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.

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