A227327 -id:A227327 - cp's OEIS frontend

A234350 Triangle T(n, k) = Number of non-equivalent (mod D_3) ways to arrange k indistinguishable points on a triangular grid of side n so that no three points are collinear. Triangle read by rows.

1, 1, 1, 1, 2, 4, 5, 2, 3, 10, 22, 24, 8, 1, 4, 22, 77, 153, 140, 47, 2, 5, 41, 217, 713, 1290, 1112, 322, 15, 7, 72, 530, 2557, 7374, 11743, 8783, 2412, 143, 1, 8, 116, 1149, 7661, 32477, 82988, 116154, 77690, 19621, 1220, 5, 10, 180, 2288, 20055, 116420, 433372

Offset: 1

Heinrich Ludwig, Dec 24 2013

The triangle T(n, k) is irregularly shaped: 1 <= k <= A234349(n). First row corresponds to n = 1.
The maximal number of points that can be placed on a triangular grid of side n so that no three points are collinear is given by A234349(n).
Without the restriction "non-equivalent (mod D_3)" the numbers are given by A194136.

			Triangle begins
1;
1,   1,    1;
2,   4,    5,    2;
3,  10,   22,   24,     8,     1;
4,  22,   77,  153,   140,    47,      2;
5,  41,  217,  713,  1290,  1112,    322,    15;
7,  72,  530, 2557,  7374, 11743,   8783,  2412,   143,    1;
8, 116, 1149, 7661, 32477, 82988, 116154, 77690, 19621, 1220, 5;
...
There are e.g. T(8, 11) = 5 non-equivalent ways to arrange 11 indistinguishable points (X) on a triangular grid of side 8 so that no point triple is collinear. As examples of the 5 solutions the 2 symmetrical ones are shown.
          .                    .
         . .                  . .
        . X .                . X .
       X . . X              X . . X
      X . . . X            . X . X .
     . . X X . .          X . . . . X
    . X . . . X .        . . X . X . .
   . . X . . X . .      . . X . . X . .

Heinrich Ludwig, Table of n, a(n) for n = 1..152

Cf. A194136
Row lengths are given by A234349
Column 1 is A001399
Column 2 is A227327 for n >= 2
Column 3 is A234351
Column 4 is A234352
Column 5 is A234353
Column 6 is A234354.

A243141 Triangle T(n, k) = Numbers of inequivalent (mod D_3) ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle of any orientation. Triangle read by rows.

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 1, 3, 10, 19, 22, 7, 1, 4, 22, 75, 170, 204, 115, 18, 1, 5, 41, 218, 816, 1891, 2635, 1909, 628, 58, 3, 7, 72, 542, 2947, 10846, 26695, 41770, 39218, 19905, 4776, 437, 13, 8, 116, 1178, 8765, 46068, 171700, 444117, 776276, 876012, 601078, 229941

Offset: 1

Heinrich Ludwig, May 30 2014

The triangle T(n, k) is irregularly shaped: 1 <= k <= A240114(n). First row corresponds to n = 1.

The maximal number of points that can be placed on a triangular grid of side n so that no three of them form an equilateral triangle is given by A240114(n).

			The triangle begins:
  1;
  1,  1;
  2,  4,   3,    1;
  3, 10,  19,   22,     7,     1;
  4, 22,  75,  170,   204,   115,    18,     1;
  5, 41, 218,  816,  1891,  2635,  1909,   628,    58,    3;
  7, 72, 542, 2947, 10846, 26695, 41770, 39218, 19905, 4776, 437, 13;
  ...
There is exactly T(5, 8) = 1 way to place 8 points (x) on a triangular grid of side 5 according to the definition of the sequence:
           .
          x x
         x . x
        x . . x
       x . . . x

Heinrich Ludwig, Table of n, a(n) for n = 1..129

Cf. A240114, A240439, A001399 (column 1), A227327 (column 2), A243142 (column 3), A243143 (column 4), A243144 (column 5).

A243207 Triangle T(n, k) = Numbers of inequivalent (mod D_3) ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid. Triangle read by rows.

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 1, 3, 10, 20, 25, 11, 3, 4, 22, 77, 186, 266, 221, 86, 14, 5, 41, 223, 881, 2344, 4238, 4885, 3451, 1296, 220, 7, 1, 7, 72, 552, 3146, 12907, 38640, 83107, 126701, 132236, 90214, 37128, 8235, 775, 24, 8, 116, 1196, 9264, 53307, 232861, 773930

Offset: 1

Heinrich Ludwig, Jun 01 2014

The triangle T(n, k) is irregularly shaped: 1 <= k <= A227308(n). First row corresponds to n = 1.

The maximal number of points that can be placed on a triangular grid of side n so that no three of them form an equilateral triangle with sides parallel to the grid is given by A227308(n).

			The triangle begins:
  1;
  1,  1;
  2,  4,   3,   1;
  3, 10,  20,  25,   11,    3;
  4, 22,  77, 186,  266,  221,   86,   14;
  5, 41, 223, 881, 2344, 4238, 4885, 3451, 1296, 220, 7, 1;
  ...
There is T(6, 12) = 1 way to place 12 points (x) on the grid obeying the rule in the definition of the sequence:
           .
          x x
         x . x
        x . . x
       x . . . x
      . x x x x .

Heinrich Ludwig, Table of n, a(n) for n = 1..153

Cf. A227308, A243211, A239572, A234247, A231655, A243141, A001399 (column 1), A227327 (column 2), A243208 (column 3), A243209 (column 4), A243210 (column 5).

A243208 Number of inequivalent (mod D_3) ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle with sides parallel to the grid.

Original entry on oeis.org

0, 3, 20, 77, 223, 552, 1196, 2380, 4388, 7657, 12710, 20301, 31297, 46892, 68426, 97674, 136596, 187713, 253770, 338217, 444773, 578018, 742852, 945210, 1191398, 1488949, 1845824, 2271415, 2775605, 3369930, 4066480, 4879238, 5822810, 6913947, 8170098, 9611127, 11257671

Offset: 2

Heinrich Ludwig, Jun 01 2014

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

Cf. A243207, A243212, A001399, A227327, A243209, A243210.

Drop[CoefficientList[Series[x^3*(-3 - 11*x - 17*x^2 - 13*x^3 - 14*x^4 - x^5 - 2*x^6 + x^7) / ((-1+x)^7 * (1+x)^3 * (1+x+x^2)), {x, 0, 50}], x],2] (* Vaclav Kotesovec, Jun 02 2014 *)

a(n) = (n^6 + 3*n^5 - 3*n^4 - 2*n^3 - 48*n^2 + 48*n)/288 + IF(MOD(n, 2) = 1)*(3*n^2 - 9*n - 1)/32 + IF(MOD(n, 3) = 1)*2/9.
G.f.: x^3*(-3 - 11*x - 17*x^2 - 13*x^3 - 14*x^4 - x^5 - 2*x^6 + x^7) / ((-1+x)^7 * (1+x)^3 * (1+x+x^2)). - Vaclav Kotesovec, Jun 02 2014
a(n) = 3*a(n-1) - 7*a(n-3) + 3*a(n-4) + 6*a(n-5) - 6*a(n-7) - 3*a(n-8) + 7*a(n-9) - 3*a(n-11) + a(n-12). - Vaclav Kotesovec, Jun 02 2014

A243209 Number of inequivalent (mod D_3) ways to place 4 points on a triangular grid of side n so that they are not vertices of an equilateral triangle with sides parallel to the grid.

Original entry on oeis.org

1, 25, 186, 881, 3146, 9264, 23810, 55058, 117205, 233135, 438544, 786541, 1354696, 2252202, 3630684, 5694984, 8718963, 13060515, 19184110, 27681103, 39300096, 54974216, 75861038, 103377456, 139251749, 185567453, 244828780, 320015885, 414665890, 532940080

Offset: 3

Heinrich Ludwig, Jun 01 2014

Heinrich Ludwig, Table of n, a(n) for n = 3..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. A243207, A243213, A001399, A227327, A243208, A243210.

Drop[CoefficientList[Series[-x^3*(1 + 22*x + 111*x^2 + 329*x^3 + 653*x^4 + 936*x^5 + 1146*x^6 + 1200*x^7 + 1150*x^8 + 900*x^9 + 650*x^10 + 286*x^11 + 131*x^12 + 28*x^13 + 19*x^14 - 5*x^15 + 3*x^16) / ((-1+x)^9 * (1+x)^4 * (1-x+x^2) * (1+x+x^2)^3), {x, 0, 40}], x],3] (* Vaclav Kotesovec, Jun 02 2014 *)

a(n) = (n^8 + 4*n^7 - 6*n^6 - 80*n^5 + 60*n^4 + 208*n^3 + 464*n^2 - 1152*n)/2304 + IF(MOD(n, 2) = 1)*(28*n^3 - 206*n^2 + 312*n + 33)/768 + IF(MOD(n, 3) = 1)*(n^2 - 2*n + 4)/18 + IF(MOD(n, 6) = 1)*(- 1/6).
G.f.: -x^3*(1 + 22*x + 111*x^2 + 329*x^3 + 653*x^4 + 936*x^5 + 1146*x^6 + 1200*x^7 + 1150*x^8 + 900*x^9 + 650*x^10 + 286*x^11 + 131*x^12 + 28*x^13 + 19*x^14 - 5*x^15 + 3*x^16) / ((-1+x)^9 * (1+x)^4 * (1-x+x^2) * (1+x+x^2)^3). - Vaclav Kotesovec, Jun 02 2014
a(n) = 3*a(n-1) - 6*a(n-3) + 6*a(n-5) + 8*a(n-6) - 12*a(n-7) - 9*a(n-8) + 13*a(n-9) + 6*a(n-10) - 6*a(n-11) - 13*a(n-12) + 9*a(n-13) + 12*a(n-14) - 8*a(n-15) - 6*a(n-16) + 6*a(n-18) - 3*a(n-20) + a(n-21). - Vaclav Kotesovec, Jun 02 2014

A230723 Number of non-equivalent ways to choose three points in an equilateral triangle grid of side n.

Original entry on oeis.org

0, 1, 6, 25, 87, 238, 575, 1228, 2425, 4446, 7734, 12806, 20422, 31444, 47072, 68639, 97929, 136893, 188061, 254170, 338679, 445297, 578616, 743524, 945968, 1192243, 1489894, 1846869, 2272575, 2776880, 3371335, 4068016, 4880921, 5824640, 6915942, 8172258, 9613470

Offset: 1

Heinrich Ludwig, Oct 28 2013

			for n = 3 there are the following a(3) = 6 choices of 3 points (=X) (rotations and reflections ignored):
    X         .         .         X         .         X
   . .       X X       . .       X 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,-7,3,6,0,-6,-3,7,0,-3,1)

Cf. A001399, A014409, A082966, A227327.

LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{0,1,6,25,87,238,575,1228,2425,4446,7734,12806},40] (* Harvey P. Dale, Oct 24 2020 *)

a(n) = (n^6 + 3*n^5 - 3*n^4 + 10*n^3 + B + C)/288
where
   B = 27*n^2 + 3*n - 9  if n odd
   B = 48*n              otherwise
and
   C = -32   if n == 1 (mod 3)
   C = 0     otherwise
G.f.: x^2*(1 + 3*x + 7*x^2 + 19*x^3 + 16*x^4 + 12*x^5 + x^6 + 2*x^7 - x^8)/((1-x^3) * (1-x^2)^3 * (1-x)^3). - Ralf Stephan, Nov 03 2013

A231653 Number of non-equivalent ways to choose 4 points in an equilateral triangle grid of side n.

Original entry on oeis.org

0, 0, 4, 41, 244, 1029, 3485, 9926, 25030, 57126, 120570, 238330, 446344, 797825, 1370684, 2274259, 3660612, 5734776, 8771181, 13127940, 19270240, 27789713, 39435814, 55142010, 76066910, 103627784, 139554142, 185929971, 245260890, 320527585, 415268815

Offset: 1

Heinrich Ludwig, Nov 12 2013

			For n = 3 there are the following a(3) = 4 choices of 4 points (=X) (rotations and reflections ignored):
    X        .        X        X
   X 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 (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1).

Cf. A234249, A001399, A227327, A230723, A231654, A231655.

a(n) = (n^8 + 4*n^7 - 6*n^6 - 32*n^5 + 84*n^4 - 32*n^3 - 16*n^2 - 192*n + B + C)/2304
where
  B = 84*n^3 - 234*n^2 + 168*n + 171   if n==1 (mod 2)
    = 0                                otherwise
and
  C = 128*n^2 + 128*n - 256            if n==1 (mod 3)
    = 0                                otherwise
G.f.: -x^3*(x^14 +7*x^12 +26*x^11 +146*x^10 +432*x^9 +947*x^8 +1418*x^7 +1621*x^6 +1405*x^5 +932*x^4 +438*x^3 +150*x^2 +33*x +4) / ((x -1)^9*(x +1)^4*(x^2 +x +1)^3). - Colin Barker, Feb 15 2014

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.

Original entry on oeis.org

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.

A243142 Number of inequivalent (mod D_3) ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.

Original entry on oeis.org

0, 3, 19, 75, 218, 542, 1178, 2350, 4340, 7585, 12605, 20153, 31094, 46620, 68068, 97212, 136008, 186975, 252855, 337095, 443410, 576378, 740894, 942890, 1188668, 1485757, 1842113, 2267125, 2770670, 3364280, 4060040, 4871928, 5814544, 6904635, 8159643, 9599427

Offset: 2

Heinrich Ludwig, May 30 2014

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,-8,14,0,-14,8,3,-4,1).

Cf. A243141, A240440, A001399, A227327, A243143, A243144.

Drop[CoefficientList[Series[x^3*(2*x^5-5*x^4+x^3-8*x^2-7*x-3) / ((x-1)^7*(x+1)^3), {x, 0, 40}], x],2] (* Vaclav Kotesovec, May 31 2014 after Colin Barker *)

concat(0, Vec(x^3*(2*x^5-5*x^4+x^3-8*x^2-7*x-3)/((x-1)^7*(x+1)^3) + O(x^100))) \\ Colin Barker, May 30 2014

a(n) = (n^6 + 3*n^5 - 5*n^4 + 6*n^3 - 68*n^2 + 72*n + IF(MOD(n, 2) = 1)*(27*n^2 - 81*n + 45))/288.

G.f.: x^3*(2*x^5-5*x^4+x^3-8*x^2-7*x-3) / ((x-1)^7*(x+1)^3). - Colin Barker, May 30 2014

A243143 Number of inequivalent (mod D_3) ways to place 4 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.

Original entry on oeis.org

1, 22, 170, 816, 2947, 8765, 22703, 52823, 113042, 225817, 426299, 766905, 1324282, 2206478, 3563770, 5599258, 8584775, 12875840, 18934040, 27347390, 38860741, 54402707, 75125825, 102441321, 138070912, 184090795, 242997153, 317760863, 411908932, 529591532, 675681764

Offset: 3

Heinrich Ludwig, May 30 2014

Heinrich Ludwig, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (5,-7,-5,23,-19,-7,27,-27,7,19,-23,5,7,-5,1).

Cf. A243141, A240441, A001399, A227327, A243142, A243144.

Drop[CoefficientList[Series[-x^3*(3*x^10 - 10*x^9 + 19*x^8 - 13*x^7 + 102*x^6 + 105*x^5 + 144*x^4 + 125*x^3 + 67*x^2 + 17*x + 1) / ((x-1)^9*(x+1)^4*(x^2+1)), {x, 0, 40}], x],3] (* Vaclav Kotesovec, May 31 2014 after Colin Barker *)

a(n) = (n^8 + 4*n^7 - 14*n^6 - 56*n^5 + 136*n^4 - 104*n^3 + 552*n^2 - 672*n)/2304 + IF(MOD(n, 2) = 1)*(28*n^3 - 198*n^2 + 296*n + 21)/768 + IF(MOD(n-1, 4) <= 1)*(-1/8).

G.f.: -x^3*(3*x^10 -10*x^9 +19*x^8 -13*x^7 +102*x^6 +105*x^5 +144*x^4 +125*x^3 +67*x^2 +17*x +1) / ((x -1)^9*(x +1)^4*(x^2 +1)). - Colin Barker, May 30 2014

cp's OEIS Frontend

A234350 Triangle T(n, k) = Number of non-equivalent (mod D_3) ways to arrange k indistinguishable points on a triangular grid of side n so that no three points are collinear. Triangle read by rows.

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A243141 Triangle T(n, k) = Numbers of inequivalent (mod D_3) ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle of any orientation. Triangle read by rows.

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A243207 Triangle T(n, k) = Numbers of inequivalent (mod D_3) ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid. Triangle read by rows.

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A243208 Number of inequivalent (mod D_3) ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle with sides parallel to the grid.

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A243209 Number of inequivalent (mod D_3) ways to place 4 points on a triangular grid of side n so that they are not vertices of an equilateral triangle with sides parallel to the grid.

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A230723 Number of non-equivalent ways to choose three points in an equilateral triangle grid of side n.

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A231653 Number of non-equivalent ways to choose 4 points in an equilateral triangle grid of side n.

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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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A243142 Number of inequivalent (mod D_3) ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.

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A243143 Number of inequivalent (mod D_3) ways to place 4 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.