A207724 Number of n X 3 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

6, 36, 78, 189, 490, 1113, 2449, 5474, 12036, 26100, 56595, 122472, 264061, 568695, 1224190, 2633000, 5660226, 12165489, 26141850, 56165805, 120662377, 259206930, 556801480, 1196027864, 2569059663, 5518244160, 11852866905, 25459111647

Offset

1,1

Comments

Column 3 of A207729.

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Robert Israel, Maple-assisted proof of formula

Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-6,0,0,3,1,0,-1).

Formula

Empirical: a(n) = 3*a(n-1) - 2*a(n-2) + 3*a(n-3) - 6*a(n-4) + 3*a(n-7) + a(n-8) - a(n-10).

Formula confirmed by Robert Israel, Mar 05 2018 (see link).

G.f.: x*(6 + 18*x - 18*x^2 + 9*x^3 + 7*x^4 + 3*x^5 - 9*x^6 - x^7 - x^8 + x^9) / ((1 - x)*(1 + x^2 - x^3)*(1 - x - x^3)*(1 - x - 2*x^2 - x^3)). - Colin Barker, Mar 05 2018

Example

Some solutions for n=4:
  1 1 0   1 1 1   1 0 0   1 0 0   0 1 1   0 1 1   0 0 1
  1 0 1   1 1 1   1 0 0   1 1 1   0 1 1   1 1 1   1 1 1
  1 0 0   1 1 0   1 0 0   1 0 0   1 1 1   0 1 1   0 0 1
  1 1 0   1 0 0   1 1 1   1 0 0   0 0 1   0 1 1   0 0 1

Maple

Configs:= select(A -> A[1..3] <> [0, 0, 0] and A[4..6] <> [0, 0, 0] and A[1..3] <> [0, 1, 0] and A[4..6] <> [0, 1, 0],
[seq(convert(x, base, 2)[1..6], x=2^6..2^7-1)]):
compat:= proc(i, j) local k, col;
   if Configs[i][4..6] <> Configs[j][1..3] then return 0 fi;
   for k from 1 to 3 do
      col:= [Configs[i][k], Configs[i][k+3], Configs[j][k+3]];
      if col = [0, 1, 1] or col = [1, 0, 1] then return 0 fi;
   od;
   1
end proc:
T:= Matrix(36, 36, compat):
u:= Vector[row](36, 1):
v:= Vector(36, 1):
6, seq(u . T^(n-2) . v, n=2..50); # Robert Israel, Mar 05 2018

Mathematica

LinearRecurrence[{3, -2, 3, -6, 0, 0, 3, 1, 0, -1}, {6, 36, 78, 189, 490, 1113, 2449, 5474, 12036, 26100}, 30] (* Jean-François Alcover, May 15 2023, after Robert Israel's confirmed formula *)

Crossrefs

Cf. A207729.

Sequence in context: A044108 A207730 A208103 * A208689 A207483 A207264

Adjacent sequences: A207721 A207722 A207723 * A207725 A207726 A207727

Keyword

nonn,easy

Author

R. H. Hardin, Feb 19 2012

Status

approved