A197237 Number of n X 3 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 4,3,2,1,2 for x=0,1,2,3,4.

1, 5, 15, 46, 156, 507, 1637, 5338, 17401, 56648, 184384, 600287, 1954546, 6363740, 20718710, 67455328, 219621081, 715042212, 2328028685, 7579574414, 24677521267, 80344901649, 261586345777, 851670870944, 2772863697333, 9027869180467

Offset

1,2

Comments

Every 0 is next to 0 4's, every 1 is next to 1 3's, every 2 is next to 2 2's, every 3 is next to 3 1's, every 4 is next to 4 2's.

Column 3 of A197242.

Links

R. H. Hardin, Table of n, a(n) for n = 1..200

Robert Israel, Maple-assisted proof of empirical formula

Formula

Empirical: a(n) = 2*a(n-1) + 8*a(n-3) + 11*a(n-4) + 17*a(n-5) + 11*a(n-6) + 5*a(n-7) - 6*a(n-8) - 16*a(n-9) - 15*a(n-10) - 3*a(n-11) + a(n-13).

Empirical formula verified (see link). - Robert Israel, Sep 26 2018

Example

Some solutions for n=4:
  1 3 1    0 0 0    2 2 2    0 0 1    0 0 0    0 0 0    1 3 1
  0 1 0    0 1 0    2 0 2    1 1 3    1 3 1    0 1 0    0 1 0
  0 1 0    1 3 1    2 0 2    3 1 1    0 1 0    0 3 1    0 2 2
  1 3 1    0 0 0    2 2 2    1 0 0    0 0 0    0 1 0    0 2 2

Maple

f:= gfun:-rectoproc({a(n) = 2*a(n-1) +8*a(n-3) +11*a(n-4) +17*a(n-5) +11*a(n-6) +5*a(n-7) -6*a(n-8) -16*a(n-9) -15*a(n-10) -3*a(n-11) +a(n-13), seq(a(i)=[1, 5, 15, 46, 156, 507, 1637, 5338, 17401, 56648, 184384, 600287, 1954546][i], i=1..13)}, a(n), remember):
map(f, [$1..30]); # Robert Israel, Sep 26 2018

Crossrefs

Sequence in context: A079798 A344814 A037504 * A307242 A296545 A089040

Adjacent sequences: A197234 A197235 A197236 * A197238 A197239 A197240

Keyword

nonn

Author

R. H. Hardin, Oct 12 2011

Status

approved