cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Heinrich Ludwig, May 11 2017

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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

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

Views

Author

Heinrich Ludwig, May 16 2017

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

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

Views

Author

Heinrich Ludwig, May 12 2017

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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

Views

Author

Heinrich Ludwig, May 12 2017

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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
Showing 1-4 of 4 results.

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