A382777 -id:A382777 - cp's OEIS frontend

A304564 Number of minimum total dominating sets in the n-triangular honeycomb bishop graph.

0, 2, 2, 6, 75, 21, 208, 3950, 540, 11220, 314880, 25740, 917280, 36029700, 1965600, 107100000, 5627890800, 219769200, 16995484800, 1153034190000, 33844456800, 3525796058400, 300234909744000, 6868433880000, 927359072640000, 96883959332160000, 1776393899280000, 301733192320560000

Offset: 1

Eric W. Weisstein, May 14 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Bishop Graph.

Cf. A304553, A304558, A382776, A382777.

T(n, k)=binomial(2*n-k, k)*binomial(n+k, n-k)*(2*(n-k))!*(2*k)!/(2^n)
b1(n) = sum(k=0, n, T(n,k))
b2(n) = sum(k=0, n, T(n,k)*(2*binomial(n+k+3,3)*(2*n-k+1) + 4*binomial(n+k+2,2)*binomial(2*n-k+2,2)))
b3(n) = sum(k=0, n, T(n,k)*(n+k)*(n+k+1)*(7*n-2*k+5)/3)
b4(n) = sum(k=0, n, T(n,k)*(2*binomial(n+k+4,4)*(2*n-k+1) + 24*binomial(n+k+2,2)*binomial(2*n-k+3,3)))
b5(n) = sum(k=0, n, T(n,k)*(40*binomial(n+k+6,6)*binomial(2*n-k+2,2) + 240*binomial(n+k+5,5)*binomial(2*n-k+3,3) + 304*binomial(n+k+4,4)*binomial(2*n-k+4,4)))
a(n) = my(t=n\3); if(n%3==0, b1(t), if(n%3==1, b2(t-1), b1(t+1) + b3(t) + b4(t-1) + b5(t-2))) \\ Andrew Howroyd, Apr 09 2025

From Andrew Howroyd, Apr 04 2025: (Start)
a(3*n) = A382777(n).
a(3*n+4) = Sum_{k=0..n} A382776(n,k)*(4*binomial(n+k+2,2) * binomial(2*n-k+2,2) + 2*binomial(n+k+3,3) * (2*n-k+1)).
See the PARI program for a(3*n+2). (End)

a(8)-a(10) from Andrew Howroyd, May 19 2018

a(11) onwards from Andrew Howroyd, May 16 2025

A382776 Triangle read by rows: T(n,k) is the number of ways to place 2n rooks on a (n+k) X (2n-k) board so that there is at least one rook in every column and row and so that each rook is defended by another.

Original entry on oeis.org

1, 1, 1, 6, 9, 6, 90, 180, 180, 90, 2520, 6300, 8100, 6300, 2520, 113400, 340200, 529200, 529200, 340200, 113400, 7484400, 26195400, 47628000, 57153600, 47628000, 26195400, 7484400, 681080400, 2724321600, 5658206400, 7858620000, 7858620000, 5658206400, 2724321600, 681080400

Offset: 0

Andrew Howroyd, Apr 04 2025

The configurations are such that k columns will each contain 2 rooks and n-k rows will each contain 2 rooks.

			Triangle begins:
        1;
        1,        1;
        6,        9,        6;
       90,      180,      180,       90;
     2520,     6300,     8100,     6300,     2520;
   113400,   340200,   529200,   529200,   340200,   113400;
  7484400, 26195400, 47628000, 57153600, 47628000, 26195400, 7484400;
  ...
The T(2,0) = 6 configurations are:
  X X . .    X . X .    X . . X    . X X .    . X . X    . . X X
  . . X X    . X . X    . X X .    X . . X    X . X .    X X . .
The T(2,1) = 9 configurations are:
  X X .   X . X   . X X   . . X   . X .   X . .   . . X   . X .   X . .
  . . X   . X .   X . .   X X .   X . X   . X X   . . X   . X .   X . .
  . . X   . X .   X . .   . . X   . X .   X . .   X X .   X . X   . X X

Row sums are A382777.

Column k=0 is A000680.

T(n,k)=binomial(2*n-k,k)*binomial(n+k,n-k)*(2*(n-k))!*(2*k)!/(2^n)

T(n,k) = binomial(2*n-k,k)*binomial(n+k,n-k)*(2*(n-k))!*(2*k)!/(2^n).

T(n,n-k) = T(n,k).

cp's OEIS Frontend

A304564 Number of minimum total dominating sets in the n-triangular honeycomb bishop graph.

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A382776 Triangle read by rows: T(n,k) is the number of ways to place 2n rooks on a (n+k) X (2n-k) board so that there is at least one rook in every column and row and so that each rook is defended by another.

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cp's OEIS Frontend

A304564 Number of minimum total dominating sets in the n-triangular honeycomb bishop graph.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A382776 Triangle read by rows: T(n,k) is the number of ways to place 2*n rooks on a (n+k) X (2*n-k) board so that there is at least one rook in every column and row and so that each rook is defended by another.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A382776 Triangle read by rows: T(n,k) is the number of ways to place 2n rooks on a (n+k) X (2n-k) board so that there is at least one rook in every column and row and so that each rook is defended by another.