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
Keywords
Examples
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
Links
- R. H. Hardin, Table of n, a(n) for n = 1..147
- Robert Israel, Maple-assisted proof of formula
Crossrefs
Cf. A183827.
Programs
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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
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
Comments