A283113 -id:A283113 - cp's OEIS frontend

A231655 Triangle T(n, k) read by rows giving number of non-equivalent ways to choose k points in an equilateral triangle grid of side n.

1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 4, 2, 1, 1, 3, 10, 25, 41, 48, 41, 25, 10, 3, 1, 1, 4, 22, 87, 244, 526, 870, 1110, 1110, 870, 526, 244, 87, 22, 4, 1, 1, 5, 41, 238, 1029, 3450, 9147, 19524, 34104, 49231, 59038, 59038, 49231, 34104, 19524, 9147, 3450, 1029, 238

Offset: 0

Heinrich Ludwig, Nov 14 2013

Number of orbits under dihedral group D_6 of order 6. - N. J. A. Sloane, Sep 12 2019

			Triangle T(n, k) is irregularly shaped: 0 <= k <= n*(n+1)/2+1. The first row corresponds to n = 1, the first column corresponds to k = 0. Rows are palindromic.
  1,  1;
  1,  1,  1,  1;
  1,  2,  4,  6,  4,  2,  1;
  1,  3, 10, 25, 41, 48, 41, 25, 10,  3,  1;
  ...
There are T(3, 2) = 4 nonisomorphic choices of 2 points (X) in an equilateral triangle grid of side 3:
      X       .       .       X
     . .     X X     . .     X .
    . X .   . . .   X . X   . . .

Heinrich Ludwig, Table of n, a(n) for n = 0..173

Columns k=1..5 are A001399, A227327, A230723, A231653, A231654.

Cf. A054252, A283113, A289709, A326611.

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

Heinrich Ludwig, Mar 01 2017

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.

			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 . .

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

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

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 *)

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

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

A283115 Number of nonequivalent ways (mod D_3) to place 4 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, 3, 40, 242, 1038, 3504, 9998, 25158, 57410, 121023, 239148, 447552, 799764, 1373400, 2278290, 3666036, 5742396, 8781111, 13141326, 19287246, 27811906, 39463424, 55177122, 76109826, 103681214, 139618479, 186008654, 245354424, 320640264, 415401264

Offset: 1

Heinrich Ludwig, Mar 01 2017

In terms of triangular chess: Number of nonequivalent ways (mod D_3) to arrange 4 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 A193982.

			There are a(6) = 3 ways to place 4 points on an 6 X 6 X 6 grid, rotations and reflections ignored:
       .             X             .
      . X           . .           X .
     . . .         . . .         . . X
    . . X .       . . X .       . . . .
   X . . . .     . X . . .     . X . . .
  . . . X . .   . . . X . .   . . . X . .

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-6,0,6,8,-12,-9,13,6,-6,-13,9,12,-8,-6,0,6,0,-3,1).

Cf. A193982, A283113, A283114 (3 points), A283116 (5 points).

Table[(5 n^8 - 100 n^7 + 810 n^6 - 3336 n^5 + 6940 n^4 - 5120 n^3 - 4080 n^2 + 6336 n)/11520 + Boole[OddQ@ n] (4 n^3 - 38 n^2 + 144 n - 207)/768 + Boole[Mod[n, 3] == 1] (n^2 - 6 n + 8)/18 - Boole[Mod[n, 6] == 1]/6, {n, 34}] (* or *)
Rest@ CoefficientList[Series[x^6*(3 + 31 x + 122 x^2 + 330 x^3 + 630 x^4 + 920 x^5 + 1128 x^6 + 1224 x^7 + 1124 x^8 + 924 x^9 + 644 x^10 + 336 x^11 + 117 x^12 + 27 x^13)/((1 - x)^9*(1 + x)^4*(1 - x + x^2) (1 + x + x^2)^3), {x, 0, 34}], x] (* Michael De Vlieger, Mar 01 2017 *)

concat(vector(5), Vec(x^6*(3 + 31*x + 122*x^2 + 330*x^3 + 630*x^4 + 920*x^5 + 1128*x^6 + 1224*x^7 + 1124*x^8 + 924*x^9 + 644*x^10 + 336*x^11 + 117*x^12 + 27*x^13) / ((1 - x)^9*(1 + x)^4*(1 - x + x^2)*(1 + x + x^2)^3) + O(x^40))) \\ Colin Barker, Mar 01 2017

a(n) = (5*n^8 - 100*n^7 + 810*n^6 - 3336*n^5 + 6940*n^4 - 5120*n^3 - 4080*n^2 + 6336*n)/11520 + IF(MOD(n, 2) = 1, 4*n^3 - 38*n^2 + 144*n - 207)/768 + IF(MOD(n, 3) = 1, n^2 - 6*n + 8)/18 + IF(MOD(n, 6) = 1, -1)/6.

G.f.: x^6*(3 + 31*x + 122*x^2 + 330*x^3 + 630*x^4 + 920*x^5 + 1128*x^6 + 1224*x^7 + 1124*x^8 + 924*x^9 + 644*x^10 + 336*x^11 + 117*x^12 + 27*x^13) / ((1 - x)^9*(1 + x)^4*(1 - x + x^2)*(1 + x + x^2)^3). - Colin Barker, 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

Heinrich Ludwig, Mar 01 2017

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.

			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 . .

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-5,-14,30,6,-50,10,44,0,-44,-10,50,-6,-30,14,5,-5,1).

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

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 *)

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

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

A283117 Number of nonequivalent ways (mod D_3) to place rooks 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

2, 2, 4, 8, 19, 51, 169, 592, 2281, 9268, 39521, 175875, 813780, 3903533, 19367571, 99208196, 523695465, 2844708347, 15877906262, 90955375095, 534101204061

Offset: 1

R. J. Mathar, Jul 07 2017

Row sums of A283113.

Cf. A289709, A326611.

a(n) = Sum_{k=0..A004396(n)} A283113(n,k).

a(n) = (A289709(n) + 2*A326611(n) + 3*2^ceiling(n/2))/6. - Andrew Howroyd, Sep 12 2019

Name changed by Andrew Howroyd, Sep 12 2019

cp's OEIS Frontend

A231655 Triangle T(n, k) read by rows giving number of non-equivalent ways to choose k points in an equilateral triangle grid of side n.

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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.

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A283115 Number of nonequivalent ways (mod D_3) to place 4 points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal.

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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.

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A283117 Number of nonequivalent ways (mod D_3) to place rooks on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal.

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