A289709 -id:A289709 - cp's OEIS frontend

A193986 T(n,k) is the number of ways to arrange k nonattacking triangular rooks on an nXnXn triangular grid.

1, 0, 3, 0, 0, 6, 0, 0, 3, 10, 0, 0, 0, 15, 15, 0, 0, 0, 2, 45, 21, 0, 0, 0, 0, 23, 105, 28, 0, 0, 0, 0, 0, 127, 210, 36, 0, 0, 0, 0, 0, 18, 468, 378, 45, 0, 0, 0, 0, 0, 0, 233, 1352, 630, 55, 0, 0, 0, 0, 0, 0, 6, 1449, 3310, 990, 66, 0, 0, 0, 0, 0, 0, 0, 270, 6213, 7190, 1485, 78, 0, 0, 0, 0

Offset: 1

R. H. Hardin, Aug 10 2011

Empirical: minimum-n nonzero T(n,k) is at n=k+floor(k/2) and this T(k+floor(k/2),k) is A002047((k-1)/2) for k odd
Table starts
...1....0......0.......0........0........0.........0.........0........0.......0
...3....0......0.......0........0........0.........0.........0........0.......0
...6....3......0.......0........0........0.........0.........0........0.......0
..10...15......2.......0........0........0.........0.........0........0.......0
..15...45.....23.......0........0........0.........0.........0........0.......0
..21..105....127......18........0........0.........0.........0........0.......0
..28..210....468.....233........6........0.........0.........0........0.......0
..36..378...1352....1449......270........0.........0.........0........0.......0
..45..630...3310....6213.....3195......166.........0.........0........0.......0
..55..990...7190...20993....21273.....4902........28.........0........0.......0
..66.1485..14260...59943...101484....54771......4842.........0........0.......0
..78.2145..26330..150903...386052...382439....104448......2532........0.......0
..91.3003..45885..344323..1243899..1976455...1127473....140598......244.......0
.105.4095..76237..726033..3527469..8250687...8147469...2568288...120052.......0
.120.5460.121688.1434678..9035376.29309540..44813100..27060693..4373740...49620
.136.7140.187712.2685046.21297492.91705972.201616740.200826477.71690568.5227020

			Some solutions for n=5 k=3
......0..........0..........0..........0..........1..........0..........0
.....0.0........0.0........0.0........0.1........0.0........0.0........0.1
....0.1.0......0.0.1......1.0.0......0.0.0......0.0.0......1.0.0......1.0.0
...0.0.0.1....1.0.0.0....0.0.1.0....0.0.1.0....0.1.0.0....0.0.1.0....0.0.1.0
..0.0.1.0.0..0.0.0.1.0..0.1.0.0.0..1.0.0.0.0..0.0.0.1.0..0.0.0.0.1..0.0.0.0.0

R. H. Hardin, Table of n, a(n) for n = 1..551

Row sums plus 1 give A289709.
Column 1 is A000217.
Column 2 is A050534.

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.

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

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

A289883 Number of maximal independent vertex sets in the n-triangular honeycomb queen graph.

Original entry on oeis.org

1, 3, 3, 11, 23, 73, 211, 651, 2371, 7401, 30151, 107713, 456409, 1888753, 8148451, 37754791, 170459451, 842310755, 4096595827, 20984546901

Offset: 1

Eric W. Weisstein, Jul 14 2017

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

Cf. A289709, A289893.

a(14)-a(20) from Andrew Howroyd, Dec 03 2021

A289708 Number of matchings in the n-triangular honeycomb queen graph.

Original entry on oeis.org

1, 4, 51, 2468, 516950, 514413280, 2620954569792

Offset: 1

Eric W. Weisstein, Jul 14 2017

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

Cf. A011848 (matching number), A289878, A289884, A289709.

Name corrected by Andrew Howroyd, Jul 17 2017

cp's OEIS Frontend

A193986 T(n,k) is the number of ways to arrange k nonattacking triangular rooks on an nXnXn triangular grid.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Extensions

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.

Views

Author

Keywords

Crossrefs

Formula

Extensions

A289883 Number of maximal independent vertex sets in the n-triangular honeycomb queen graph.

Views

Author

Keywords

Links

Crossrefs

Extensions

A289708 Number of matchings in the n-triangular honeycomb queen graph.

Views

Author

Keywords

Links

Crossrefs

Extensions