A350336 Number of n X n ternary matrices with no two adjacent 0's.

1, 3, 56, 7504, 6832640, 42780151808, 1836366011301888, 540795841280638713856, 1092417949346109029345132544, 15137179876232766647722798101823488, 1438787206346713875314130065804001328234496, 938091111277955250977701268973340995182098116509696

Offset: 0

Author

Robert P. P. McKone, Jan 03 2022

Keywords

• nonn
• hard

Comments

A two-dimensional generalization of A028859.

2^(n^2) < a(n) < 3^(n^2).

Examples

			a(1) is trivial because all 3 1 X 1 matrices have no 2 adjacent 0's, whereas for a(2) the 56 matrices are:
  {
    {{0, 1}, {1, 0}}, {{0, 1}, {1, 1}},
    {{0, 1}, {1, 2}}, {{0, 1}, {2, 0}},
    {{0, 1}, {2, 1}}, {{0, 1}, {2, 2}},
    {{0, 2}, {1, 0}}, {{0, 2}, {1, 1}},
    {{0, 2}, {1, 2}}, {{0, 2}, {2, 0}},
    {{0, 2}, {2, 1}}, {{0, 2}, {2, 2}},
    {{1, 0}, {0, 1}}, {{1, 0}, {0, 2}},
    {{1, 0}, {1, 1}}, {{1, 0}, {1, 2}},
    {{1, 0}, {2, 1}}, {{1, 0}, {2, 2}},
    {{1, 1}, {0, 1}}, {{1, 1}, {0, 2}},
    {{1, 1}, {1, 0}}, {{1, 1}, {1, 1}},
    {{1, 1}, {1, 2}}, {{1, 1}, {2, 0}},
    {{1, 1}, {2, 1}}, {{1, 1}, {2, 2}},
    {{1, 2}, {0, 1}}, {{1, 2}, {0, 2}},
    {{1, 2}, {1, 0}}, {{1, 2}, {1, 1}},
    {{1, 2}, {1, 2}}, {{1, 2}, {2, 0}},
    {{1, 2}, {2, 1}}, {{1, 2}, {2, 2}},
    {{2, 0}, {0, 1}}, {{2, 0}, {0, 2}},
    {{2, 0}, {1, 1}}, {{2, 0}, {1, 2}},
    {{2, 0}, {2, 1}}, {{2, 0}, {2, 2}},
    {{2, 1}, {0, 1}}, {{2, 1}, {0, 2}},
    {{2, 1}, {1, 0}}, {{2, 1}, {1, 1}},
    {{2, 1}, {1, 2}}, {{2, 1}, {2, 0}},
    {{2, 1}, {2, 1}}, {{2, 1}, {2, 2}},
    {{2, 2}, {0, 1}}, {{2, 2}, {0, 2}},
    {{2, 2}, {1, 0}}, {{2, 2}, {1, 1}},
    {{2, 2}, {1, 2}}, {{2, 2}, {2, 0}},
    {{2, 2}, {2, 1}}, {{2, 2}, {2, 2}}
  }
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..30

Crossrefs

Cf. A006506 for binary version.

Cf. A028859 for one-dimensional version.

Programs

Mathematica

t[m_] := t[m] = Map[ArrayReshape[#, {m, m}] &, Tuples[{0, 1, 2}, m^2]];a[m_] := a[m] = Count[Table[AnyTrue[Flatten[{Table[Equal[0, t[m][[n, a, b]], t[m][[n, a, b + 1]]], {a, 1, m}, {b, 1, m - 1}], Table[Equal[0, t[m][[n, a, b]], t[m][[n, a + 1, b]]], {a, 1, m - 1}, {b, 1, m}]}], TrueQ], {n, 1, 3^(m^2)}], False]; Table[a[n], {n, 1, 3}]

Extensions

Terms a(5)-a(11) from Andrew Howroyd, Jan 04 2022