A326611 -id:A326611 - cp's OEIS frontend

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

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

A231655 Triangle T(n, k) read by rows giving number of non-equivalent ways to choose k points in an equilateral triangle grid of side n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 4, 2, 1, 1, 3, 10, 25, 41, 48, 41, 25, 10, 3, 1, 1, 4, 22, 87, 244, 526, 870, 1110, 1110, 870, 526, 244, 87, 22, 4, 1, 1, 5, 41, 238, 1029, 3450, 9147, 19524, 34104, 49231, 59038, 59038, 49231, 34104, 19524, 9147, 3450, 1029, 238

Offset: 0

Heinrich Ludwig, Nov 14 2013

Number of orbits under dihedral group D_6 of order 6. - N. J. A. Sloane, Sep 12 2019

			Triangle T(n, k) is irregularly shaped: 0 <= k <= n*(n+1)/2+1. The first row corresponds to n = 1, the first column corresponds to k = 0. Rows are palindromic.
  1,  1;
  1,  1,  1,  1;
  1,  2,  4,  6,  4,  2,  1;
  1,  3, 10, 25, 41, 48, 41, 25, 10,  3,  1;
  ...
There are T(3, 2) = 4 nonisomorphic choices of 2 points (X) in an equilateral triangle grid of side 3:
      X       .       .       X
     . .     X X     . .     X .
    . X .   . . .   X . X   . . .

Heinrich Ludwig, Table of n, a(n) for n = 0..173

Columns k=1..5 are A001399, A227327, A230723, A231653, A231654.

Cf. A054252, A283113, A289709, A326611.

A283117 Number of nonequivalent ways (mod D_3) to place rooks on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal.

Original entry on oeis.org

2, 2, 4, 8, 19, 51, 169, 592, 2281, 9268, 39521, 175875, 813780, 3903533, 19367571, 99208196, 523695465, 2844708347, 15877906262, 90955375095, 534101204061

Offset: 1

R. J. Mathar, Jul 07 2017

Row sums of A283113.

Cf. A289709, A326611.

a(n) = Sum_{k=0..A004396(n)} A283113(n,k).

a(n) = (A289709(n) + 2*A326611(n) + 3*2^ceiling(n/2))/6. - Andrew Howroyd, Sep 12 2019

Name changed by Andrew Howroyd, Sep 12 2019

cp's OEIS Frontend

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

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A231655 Triangle T(n, k) read by rows giving number of non-equivalent ways to choose k points in an equilateral triangle grid of side n.

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A283117 Number of nonequivalent ways (mod D_3) to place rooks on an n X n X n triangular grid so that no two of them are on the same row, column or diagonal.

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