A240439 -id:A240439 - cp's OEIS frontend

A240440 Number of 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.

0, 0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, 45045, 75075, 120120, 185640, 278460, 406980, 581400, 813960, 1119195, 1514205, 2018940, 2656500, 3453450, 4440150, 5651100, 7125300, 8906625, 11044215, 13592880, 16613520, 20173560, 24347400, 29216880

Offset: 1

Heinrich Ludwig, Apr 08 2014

a(n) = 15 * A000579(n+3).

a(n) = A001498(n,3), the fourth column of coefficients of Bessel polynomials. - Ran Pan, Dec 03 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. (2010) Vol. 13, Issue 4, Art. No. 10.4.4. See p=5 in the last equation on page 3.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000579, A001498, A240441, A240442, A240439.

If one of the initial zeros is omitted, this is a row of the array in A129533.

[(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48 : n in [1..50]]; // Wesley Ivan Hurt, Dec 03 2015

A240440:=n->(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48; seq(A240440(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2014

Table[(n+3)(n+2)(n+1)n(n-1)(n-2)/48, {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[15 x^2/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)

Vec(15*x^3/(1-x)^7 + O(x^100)) \\ Colin Barker, Apr 18 2014

vector(100,n,(n^2-1)*(n^2-4)*(n+3)*n/48) \\ Derek Orr, Dec 24 2015

a(n) = (n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48.
G.f.: 15*x^3 / (1-x)^7. - Colin Barker, Apr 18 2014
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7) for n>7. - Wesley Ivan Hurt, Dec 03 2015

A240441 Number of ways to place 4 points on a triangular grid of side n so that no three of these points are vertices of an equilateral triangle of any orientation.

Original entry on oeis.org

0, 0, 3, 114, 969, 4773, 17415, 52125, 135375, 315675, 676200, 1352085, 2553558, 4595934, 7937874, 13229118, 21369330, 33579450, 51487425, 77229900, 113571975, 164046795, 233117313, 326362179, 450688329, 614572413, 828333870, 1104441975, 1457859900, 1906428300

Offset: 1

Heinrich Ludwig, Apr 05 2014

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-19,21,6,-42,42,-6,-21,19,-7,1).

Cf. A240439, A240440, A240442.

Table[(n^8+4*n^7-14*n^6-56*n^5+61*n^4+220*n^3-84*n^2-240*n)/384 +If[EvenQ[n],0,(6*n+3)/32],{n,1,20}] (* Vaclav Kotesovec, Apr 05 2014 after Heinrich Ludwig *)

concat([0,0], Vec(-3*x^3*(x^4+31*x^3+76*x^2+31*x+1)/((x-1)^9*(x+1)^2) + O(x^100))) \\ Colin Barker, Apr 05 2014

a(n) = (n^8 + 4*n^7 - 14*n^6 - 56*n^5 + 61*n^4 + 220*n^3 - 84*n^2 - 240*n)/384 + IF(MOD(n, 2) = 1)*(6*n + 3)/32.

G.f.: -3*x^3*(x^4+31*x^3+76*x^2+31*x+1) / ((x-1)^9*(x+1)^2). - Colin Barker, Apr 05 2014

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).

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).

A240442 Number of ways to place 5 points on a triangular grid of side n so that no three of these points are vertices of an equilateral triangle of any orientation.

Original entry on oeis.org

0, 39, 1194, 11259, 64776, 275805, 957516, 2859768, 7606821, 18444537, 41458599, 87464157, 174846963, 333687378, 611613150, 1081890447

Offset: 3

Heinrich Ludwig, Apr 08 2014

Cf. A240439, A240440, A240441.

a(n) = (n^10 + 5*n^9 - 30*n^8 - 150*n^7)/3840 + O(n^6).

a(16)-a(18) from Heinrich Ludwig, Apr 25 2014

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A240440 Number of 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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A240441 Number of ways to place 4 points on a triangular grid of side n so that no three of these points are vertices of an equilateral triangle of any orientation.

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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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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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A240442 Number of ways to place 5 points on a triangular grid of side n so that no three of these points are vertices of an equilateral triangle of any orientation.

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