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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. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

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

EXAMPLE

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.

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

Sequence in context: A319699 A157636 A086626 * A300695 A296186 A232967

Adjacent sequences:  A244497 A244498 A244499 * A244501 A244502 A244503

KEYWORD

nonn,tabf

AUTHOR

Heinrich Ludwig, Jun 29 2014

STATUS

approved

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Last modified November 14 20:11 EST 2019. Contains 329129 sequences. (Running on oeis4.)