A183821 1/256 the number of (n+1) X 3 0..3 arrays with no 2 X 2 subblock being a reflection across the shared element pair of any horizontal or vertical neighbor.

15, 813, 43947, 2377341, 128578815, 6954559893, 376152548283, 20345104031589, 1100412240703119, 59518368554767389, 3219189893127084843, 174117402720326269485, 9417546276713264441151, 509369980225226329837125

Offset

1,1

Comments

Column 2 of A183827.

Links

R. H. Hardin, Table of n, a(n) for n = 1..147

Robert Israel, Maple-assisted proof of formula

Formula

Empirical: a(n) = 43*a(n-1) + 645*a(n-2) - 2451*a(n-3).

Empirical g.f.: 3*x*(5 + 56*x - 229*x^2) / (1 - 43*x - 645*x^2 + 2451*x^3). - Colin Barker, Apr 05 2018

Empirical formulas verified (see link). - Robert Israel, Aug 10 2018

Example

Some solutions with upper left block zero for 5 X 3:
   0 0 2   0 0 0   0 0 3   0 0 3   0 0 0   0 0 2   0 0 0
   0 0 0   0 0 1   0 0 3   0 0 2   0 0 1   0 0 1   0 0 2
   2 0 0   2 0 1   3 0 2   1 0 2   1 0 3   1 0 0   3 0 1
   2 2 1   3 2 3   1 3 1   1 3 1   0 3 3   2 1 2   0 3 3
   3 2 3   0 0 3   0 0 3   2 0 3   3 2 2   0 0 3   2 2 0

Maple

Configs:= remove(t -> [t[1], t[4]]=[t[3], t[6]], [seq(convert(4^6+i, base, 4)[1..6], i=0..4^6-1)]): nConfigs:= nops(Configs):
Compatible:= proc(i, j) local k;
if Configs[i][4..6] <> Configs[j][1..3] or Configs[i][1..2] =Configs[j][4..5] or Configs[i][2..3] = Configs[j][5..6] then 0 else 1 fi;
end proc:
T:= Matrix(nConfigs, nConfigs, Compatible):
u:= Vector(nConfigs, 1):
Tu[0]:= u:
for n from 1 to 30 do Tu[n]:= T . Tu[n-1] od:
seq(u^%T . Tu[n]/256, n=0..30); # Robert Israel, Aug 10 2018

Crossrefs

Cf. A183827.

Sequence in context: A280310 A211104 A279493 * A301312 A374581 A230181

Adjacent sequences: A183818 A183819 A183820 * A183822 A183823 A183824

Keyword

nonn

Author

R. H. Hardin, Jan 07 2011

Status

approved