A287204 -id:A287204 - 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

A287230 Number of matchings in the n-triangular honeycomb acute knight graph.

Original entry on oeis.org

1, 1, 8, 64, 1331, 64000, 6400075, 1404928000, 677298787768, 712186032947200, 1635557819719974912, 8209592592625295700893, 90036881979773511965369428, 2157454308508779392217680572439, 112955975573487831948842897960075264, 12921763288870998051759383983484279183072, 3229803978189426975602886931834056243712000000

Offset: 1

Eric W. Weisstein, May 22 2017

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, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching

Cf. A289143, A287204, A289904, A287228, A269869.

a(11) from Eric W. Weisstein, Jun 25 2017
a(12)-a(14) from Andrew Howroyd, Jul 17 2017
a(15)-a(17) from Eric W. Weisstein, Sep 02 2025

A289902 Number of dominating sets in the n-triangular honeycomb acute knight graph.

Original entry on oeis.org

1, 1, 27, 441, 9261, 421875, 47177249, 10546683057, 4466853289709, 3723323714676297, 6207276939337266129, 20676801823320497569317, 136896643642841500100918369

Offset: 1

Eric W. Weisstein, Jul 14 2017

Eric Weisstein's World of Mathematics, Dominating Set

Cf. A289903, A287204, A287064.

a(8) from Andrew Howroyd, Jul 17 2017

a(9)-a(13) from Eric W. Weisstein, Feb 09 2024

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A287230 Number of matchings in the n-triangular honeycomb acute knight graph.

Views

Author

Keywords

Links

Crossrefs

Extensions

A289902 Number of dominating sets in the n-triangular honeycomb acute knight graph.

Views

Author

Keywords

Links

Crossrefs

Extensions