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

Author

Heinrich Ludwig, Jun 29 2014

Keywords

• nonn
• tabf

Comments

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

Examples

			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.
		

Links

• 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

Crossrefs

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