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 1-3 of 3 results.

A283113 Triangle read by rows: T(n,k) is the number of nonequivalent ways (mod D_3) to place k points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal (n >= 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 9, 5, 1, 5, 19, 23, 3, 1, 7, 38, 82, 40, 1, 1, 8, 66, 230, 242, 45, 1, 10, 110, 560, 1038, 533, 29, 1, 12, 170, 1208, 3504, 3546, 821, 6, 1, 14, 255, 2392, 9998, 16917, 9137, 807, 1, 16, 365, 4405, 25158, 64345, 63755, 17408, 422
Offset: 1

Views

Author

Heinrich Ludwig, Mar 10 2017

Keywords

Comments

Length of n-th row is A004396(n) + 1, for 1 <= n <= 21, where A004396(n) is the maximal number of points that can be placed under the condition mentioned above.
Rotations and reflections of placements are not counted. If they are to be counted, see A193986.
In terms or triangular chess: Number of nonequivalent ways (mod D_3) to arrange k nonattacking rooks on an n X n X n board, k>=0, n>=1.

Examples

			The table begins with T(1,0), T(1,1);
  1,  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,   1;
  1,  4,   9,   5;
  1,  5,  19,  23,    3;
  1,  7,  38,  82,   40,   1;
  1,  8,  66, 230,  242,  45;
  1, 10, 110, 560, 1038, 533, 29;
  ...

Links

Crossrefs

Row sums give A283117.
Columns 2..6: A001399, A005994, A283114, A283115, A283116.
Cf. A004396, A193986.

A283114 Number of nonequivalent ways (mod D_3) to place 3 points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal.

Original entry on oeis.org

0, 0, 0, 1, 5, 23, 82, 230, 560, 1208, 2392, 4405, 7673, 12733, 20320, 31326, 46914, 68460, 97698, 136635, 187737, 253813, 338240, 444818, 578038, 742898, 945224, 1191443, 1488955, 1845865, 2271410, 2775640, 3369910, 4066506, 4879200, 5822823, 6913887, 8170095
Offset: 1

Views

Author

Heinrich Ludwig, Mar 01 2017

Keywords

Comments

In terms of triangular chess: Number of nonequivalent ways (mod D_3) to arrange 3 nonattacking rooks on an n X n X n board.
Reflections and rotations of placements are not counted. For numbers if they are to be counted see A193981.

Examples

			There is a(4) = 1 way to place 3 points on a 4 X 4 X 4 grid, rotations and reflections ignored:
     .
    X .
   . . X
  . X . .

Links

Crossrefs

Cf. A193981, A283113, A283115 (4 points), A283116 (5 points).

Programs

  • Mathematica
    Table[(n^6 - 9 n^5 + 27 n^4 - 20 n^3 - 24 n^2 + 24 n)/288 + Boole[OddQ@ n] (n^2 - 3 n - 5)/32 + Boole[Mod[n, 3] == 1] 2/9, {n, 38}] (* or *)
Rest@ CoefficientList[Series[x^4*(1 + 2 x + 8 x^2 + 20 x^3 + 16 x^4 + 10 x^5 + 3 x^6)/((1 - x)^7*(1 + x)^3*(1 + x + x^2)), {x, 0, 38}], x] (* Michael De Vlieger, Mar 01 2017 *)
LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{0,0,0,1,5,23,82,230,560,1208,2392,4405},40] (* Harvey P. Dale, May 07 2022 *)
  • PARI
    concat(vector(3), Vec(x^4*(1 + 2*x + 8*x^2 + 20*x^3 + 16*x^4 + 10*x^5 + 3*x^6) / ((1 - x)^7*(1 + x)^3*(1 + x + x^2)) + O(x^30))) \\ Colin Barker, Mar 01 2017

Formula

a(n) = (n^6 - 9*n^5 + 27*n^4 - 20*n^3 - 24*n^2 + 24*n)/288 + IF(MOD(n, 2) = 1, n^2 - 3*n - 5)/32 + IF(MOD(n, 3) = 1, 2)/9.
G.f.: x^4*(1 + 2*x + 8*x^2 + 20*x^3 + 16*x^4 + 10*x^5 + 3*x^6) / ((1 - x)^7*(1 + x)^3*(1 + x + x^2)). - Colin Barker, Mar 01 2017
a(n) = ( 2*n^6 - 18*n^5 + 54*n^4 - 40*n^3 - 39*n^2 + 21*n - 45 - 9*(n^2 - 3*n - 5)*(-1)^n + 128*((n mod 3) mod 2) )/576. - Bruno Berselli, Mar 01 2017

A283116 Number of nonequivalent ways (mod D_3) to place 5 points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 45, 533, 3546, 16917, 64345, 207327, 587922, 1505924, 3549610, 7806420, 16188690, 31919658, 60238044, 109392480, 192015912, 326985561, 541900545, 876326275, 1385991432, 2148140345, 3268293567, 4888684275, 7198705228, 10447710630, 14960606226
Offset: 1

Views

Author

Heinrich Ludwig, Mar 01 2017

Keywords

Comments

In terms of triangular chess: Number of nonequivalent ways (mod D_3) to arrange 5 nonattacking rooks on an n X n X n board.
Reflections and rotations of placements are not counted. For numbers if they are to be counted see A193983.

Examples

			There is a(7) = 1 way to place 5 points on a 7 X 7 X 7 grid, rotations and reflections ignored:
         .
        . .
       . X .
      . . . X
     X . . . .
    . . X . . .
   . . . . X . .

Links

Crossrefs

Cf. A193983, A283113, A283114 (3 points), A283115 (4 points).

Programs

  • Mathematica
    Table[(n^10 - 35 n^9 + 530 n^8 - 4526 n^7 + 23693 n^6 - 76544 n^5 + 141360 n^4 - 104944 n^3 - 67984 n^2 + 124224 n)/23040 + Boole[OddQ@ n] (-45 n^4 + 1350 n^3 - 13770 n^2 + 57915 n - 81225)/23040 - 2 Boole[Mod[n, 3] == 2]/9, {n, 32}] (* or *)
Rest@ CoefficientList[Series[x^7*(1 + 40 x + 313 x^2 + 1120 x^3 + 2452 x^4 + 3596 x^5 + 3621 x^6 + 2512 x^7 + 1149 x^8 + 316 x^9)/((1 - x)^11*(1 + x)^5*(1 + x + x^2)), {x, 0, 32}], x] (* Michael De Vlieger, Mar 01 2017 *)
  • PARI
    concat(vector(6), Vec(x^7*(1 + 40*x + 313*x^2 + 1120*x^3 + 2452*x^4 + 3596*x^5 + 3621*x^6 + 2512*x^7 + 1149*x^8 + 316*x^9) / ((1 - x)^11*(1 + x)^5*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Mar 01 2017

Formula

a(n) = (n^10 - 35*n^9 + 530*n^8 - 4526*n^7 + 23693*n^6 - 76544*n^5 + 141360*n^4 - 104944*n^3 - 67984*n^2 + 124224*n)/23040 + IF(MOD(n, 2) = 1, - 45*n^4 + 1350*n^3 - 13770*n^2 + 57915*n - 81225)/23040 + IF(MOD(n, 3) = 2, -2)/9.
G.f.: x^7*(1 + 40*x + 313*x^2 + 1120*x^3 + 2452*x^4 + 3596*x^5 + 3621*x^6 + 2512*x^7 + 1149*x^8 + 316*x^9) / ((1 - x)^11*(1 + x)^5*(1 + x + x^2)). - Colin Barker, Mar 01 2017
Showing 1-3 of 3 results.

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