A283117 -id:A283117 - cp's OEIS frontend

A283113 Triangle read by rows: T(n,k) is the number of nonequivalent ways (mod D_3) to place k points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal (n >= 1).

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 9, 5, 1, 5, 19, 23, 3, 1, 7, 38, 82, 40, 1, 1, 8, 66, 230, 242, 45, 1, 10, 110, 560, 1038, 533, 29, 1, 12, 170, 1208, 3504, 3546, 821, 6, 1, 14, 255, 2392, 9998, 16917, 9137, 807, 1, 16, 365, 4405, 25158, 64345, 63755, 17408, 422

Offset: 1

Heinrich Ludwig, Mar 10 2017

Length of n-th row is A004396(n) + 1, for 1 <= n <= 21, where A004396(n) is the maximal number of points that can be placed under the condition mentioned above.
Rotations and reflections of placements are not counted. If they are to be counted, see A193986.
In terms or triangular chess: Number of nonequivalent ways (mod D_3) to arrange k nonattacking rooks on an n X n X n board, k>=0, n>=1.

			The table begins with T(1,0), T(1,1);
  1,  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,   1;
  1,  4,   9,   5;
  1,  5,  19,  23,    3;
  1,  7,  38,  82,   40,   1;
  1,  8,  66, 230,  242,  45;
  1, 10, 110, 560, 1038, 533, 29;
  ...

Heinrich Ludwig, Table of n, a(n) for n = 1..175

Row sums give A283117.
Columns 2..6: A001399, A005994, A283114, A283115, A283116.
Cf. A004396, A193986.

A289709 Number of independent vertex sets and vertex covers in the n-triangular honeycomb queen graph.

Original entry on oeis.org

2, 4, 10, 28, 84, 272, 946, 3486, 13560, 55432, 236852, 1054928, 4881972, 23420436, 116204016, 595246848, 3142169416, 17068245184, 95267426432, 545732236936, 3204607199704

Offset: 1

Eric W. Weisstein, Jul 14 2017

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

Cf. A193986, A283117, A289883, A289893, A326611.

From Andrew Howroyd, Sep 12 2019: (Start)
a(n) = 6*A283117(n) - 2*A326611(n) - 3*2^ceiling(n/2).
a(n) = 1 + Sum_{k>=1} A193986(n,k).
(End)

a(13)-a(21) from Andrew Howroyd, Sep 12 2019

A326611 Number of arrangements of rooks with rotational symmetry on a triangular grid with n grid points on each side and no two rooks on the same row, column or diagonal.

Original entry on oeis.org

2, 1, 1, 4, 3, 5, 10, 9, 15, 40, 41, 65, 162, 189, 321, 780, 919, 1681, 4034, 5281, 9259, 23936, 30665, 57601, 143602, 199577, 367561, 959236, 1323243, 2585133, 6580650, 9609145, 18433799, 49030248, 71211721, 142636377, 371147842, 566921925, 1122881889, 3024341084, 4583822647, 9446124313

Offset: 1

Andrew Howroyd, Sep 12 2019

nonn

			The four cases for n = 4 are:
         o               o               o               o
       o   o           o   o           X   o           o   X
     o   o   o       o   X   o       o   o   X       X   o   o
   o   o   o   o   o   o   o   o   o   X   o   o   o   o   X   o

Cf. A283117, A289709.

def solve(cli):
    count = 1
    for k in range(len(cli)):
        x,y,z = cli[k]
        clo = []
        for c in cli[k+1:]:
            if (not x in c) and (not y in c) and (not z in c):
                clo.append(c)
        count += 2*solve(clo)
    return count
def A326611(n):
    c0 = []
    for x in range(n):
        for y in range(x+1,n):
            z = n-1-x-y
            if z>y: c0.append((x,y,z))
    count = solve(c0)
    if n%3 == 1:
        c1 = [c for c in c0 if not n//3 in c]
        count += solve(c1)
    return count
# Bert Dobbelaere, May 14 2021

More terms from Bert Dobbelaere, May 14 2021

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A283113 Triangle read by rows: T(n,k) is the number of nonequivalent ways (mod D_3) to place k points on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal (n >= 1).

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A289709 Number of independent vertex sets and vertex covers in the n-triangular honeycomb queen graph.

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A326611 Number of arrangements of rooks with rotational symmetry on a triangular grid with n grid points on each side and no two rooks on the same row, column or diagonal.

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