A243212 -id:A243212 - cp's OEIS frontend

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.

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

A243211 Triangle T(n, k) = Numbers of 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, 3, 3, 1, 6, 15, 15, 3, 1, 10, 45, 107, 128, 63, 10, 1, 15, 105, 428, 1062, 1566, 1276, 507, 69, 1, 21, 210, 1282, 5160, 13971, 25191, 29235, 20508, 7747, 1251, 42, 1, 1, 28, 378, 3198, 18591, 77124, 231090, 498097, 759117, 792942, 540361, 222597, 49053

Offset: 1

Heinrich Ludwig, Jun 09 2014

The triangle T(n, k) is irregularly shaped: 0 <= 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,  3,   3;
  1,  6,  15,   15,    3;
  1, 10,  45,  107,  128,    63,    10,
  1, 15, 105,  428, 1062,  1566,  1276,   507,    69,
  1, 21, 210, 1282, 5160, 13971, 25191, 29235, 20508, 7747, 1251, 42, 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..165

Cf. A227308, A243207, A084546, A234251, A239567, A240439, A194136, A000217 (column 2), A050534 (column 3), A243212 (column 4), A243213 (column 5), A243214 (column 6).

A243213 Number of ways to place 4 points on a triangular grid of side length n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.

Original entry on oeis.org

3, 128, 1062, 5160, 18591, 55113, 142005, 329045, 701160, 1395975, 2626953, 4713723, 8120322, 13503350, 21770766, 34153758, 52292385, 78337890, 115072320, 166048850, 235753353, 329791143, 455099307, 620189115, 835418766, 1113301553, 1468849515, 1919958285

Offset: 3

Heinrich Ludwig, Jun 09 2014

			There are exactly a(3) = 3 ways to place 4 points (x) on a 3X3X3 grid, no three of them being vertices of an equilateral triangle:
      .            x            x
     x x          . x          x .
    x . x        x x .        . x x

Heinrich Ludwig, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,2,27,-36,0,36,-27,-2,12,-6,1)

Cf. A243211, A243209, A000217, A050534, A243212, A243214.

Vec(x^3*(7*x^7-33*x^6-15*x^5-38*x^4-318*x^3-330*x^2-110*x-3)/((x-1)^9*(x+1)^3) + O(x^100)) \\ Colin Barker, Jun 09 2014

a(n) = (n^8 + 4*n^7 - 6*n^6 - 80*n^5 - 15*n^4 + 532*n^3 - 244*n^2 - 432*n)/384 + IF(MOD(n, 2) = 1)*(-n^2 - n + 12)/16.

G.f.: x^3*(7*x^7-33*x^6-15*x^5-38*x^4-318*x^3-330*x^2-110*x-3) / ((x-1)^9*(x+1)^3). - Colin Barker, Jun 09 2014

A243214 Number of ways to place 5 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.

Original entry on oeis.org

0, 63, 1566, 13971, 77124, 319206, 1083723, 3181401, 8344854, 20006349, 44548227, 93248628, 185176866, 351410664, 640972980, 1129067352, 1928196432, 3203016813

Offset: 3

Heinrich Ludwig, Jun 10 2014

Cf. A243211, A243210, A000217, A050534, A243212, A243213.

a(n) = (n^10 + 5*n^9 - 10*n^8)/3840 + O(n^7)

cp's OEIS Frontend

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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A243211 Triangle T(n, k) = Numbers of 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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A243213 Number of ways to place 4 points on a triangular grid of side length n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.

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A243214 Number of ways to place 5 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.

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