A244507 -id:A244507 - cp's OEIS frontend

A244500 Number T(n, k) of ways to place k points on an n X n X n triangular grid so that no pair of them has distance sqrt(3). Triangle read by rows.

1, 1, 1, 3, 3, 1, 1, 6, 12, 8, 1, 10, 36, 55, 33, 9, 1, 15, 87, 248, 378, 339, 187, 63, 12, 1, 1, 21, 180, 820, 2190, 3606, 3716, 2340, 825, 125, 1, 28, 333, 2212, 9110, 24474, 43928, 53018, 42774, 22792, 7945, 1764, 196, 1, 36, 567, 5163, 30300, 121077, 339621

Offset: 1

Heinrich Ludwig, Jun 29 2014

In the following triangular grid points x have Euclidean distance sqrt(3) from point o. It is the second closest distance possible among grid points.
      x
     . .
    . o .
   x . . x
Triangle T(n, k) is irregular: 0 <= k <= max(n), where max(n), the maximal number of points that can be placed on the grid, is:
  for n = 3j-2:            max(n) = A000326(j) = j(3j-1)/2;
  for n = 3j-1 or n = 3j:  max(n) = A045943(j) = 3j(j+1)/2; j = 1,2,3,...
Empirical: (1) The number of ways to place the maximal number of points for grid sizes n = 3j are cubes of Catalan numbers, i.e., for n = 3j: T(n, max(n)) = C(j+1)^3 = A033536(j+1). (2) For n = 3j-2: T(n, max(n)) = A244506(n) = A244507^2(n). (3) For n = 3j-1: T(n, max(n)) = A000012(n) = 1 and T(n, max(n)-1) = 3j^2.
Row n is also the coefficients of the independence polynomial of the n-triangular honeycomb acute knight graph. - Eric W. Weisstein, May 21 2017

			On an 8 X 8 X 8 grid there is T(8, 18) = 1 way to place 18 points (x) so that no pair of points has the distance square root of 3.
         x
        x x
       . . .
      x . . x
     x x . x x
    . . . . . .
   x . . x . . x
  x x . x x . x x
Continuation of this pattern will give the unique maximal solution for all n = 3j-1.
Triangle T(n, k) begins:
  1,  1;
  1,  3,   3,   1;
  1,  6,  12,   8;
  1, 10,  36,  55,   33,    9;
  1, 15,  87, 248,  378,  339,  187,   63,  12,   1;
  1, 21, 180, 820, 2190, 3606, 3716, 2340, 825, 125;
First row refers to n = 1.

Heinrich Ludwig, Table of n, a(n) for n = 1..153
Stan Wagon, Graph Theory Problems from Hexagonal and Traditional Chess, The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287.
Eric Weisstein's World of Mathematics, Independence Polynomial

Cf. A000108, A000326, A033536,  A045943, A244506, A244507.
Cf. A000217 (column 2), A086274 (1/3 * column 3), A244501 (column 4), A244502 (column 5), A244503 (column 6).
Cf. A287195 (length of row n). - Eric W. Weisstein, May 21 2017
Cf. A287204 (row sums). - Eric W. Weisstein, May 21 2017

A244506 Number of ways to place the maximal number of points that can be placed on a j X j X j triangular grid, j=3n-2, so that no pair of them has distance sqrt(3).

Original entry on oeis.org

1, 9, 196, 6084, 219024, 8450649, 338265664, 13840346025, 574510941225, 24093764931600

Offset: 1

Heinrich Ludwig, Jul 11 2014

(1) All a(n) are square numbers. The sequence of their roots is A244507.
(2) On a j X j X j grid, j = 3n-2, the maximal number of points that can be placed is the pentagonal number A000326(n).
(3) On a maximally occupied grid, the following grid points "X" are always occupied (example for j = 7, for other j's expand this pattern):
              X
             . .
            . . .
           x . . X
          . . . . .
         . . . . . .
        X . . X . . X
(4) For j X j X j grids, j = 3n, the corresponding numbers are cubes of Catalan numbers, A033536(n). For j = 3n-1, the corresponding numbers are always 1, A000012(n).

Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set

Cf. A000326, A244507, A244500.

Cf. A297537 (maximum independent vertex sets for n-triangular honeycomb acute knight graph).

a(10) from Heinrich Ludwig, Aug 17 2014

cp's OEIS Frontend

A244500 Number T(n, k) of ways to place k points on an n X n X n triangular grid so that no pair of them has distance sqrt(3). Triangle read by rows.

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