A233807 Number of tilings of an n X n square using right trominoes and at most one monomino.

1, 1, 4, 0, 16, 128, 162, 34528, 943096, 1193600, 3525377600, 480585761344, 2033502499954, 46983507796973152, 32908187880881958736, 458324092996867592192, 83153202122213272708832688, 299769486068040749617049301344

Offset: 0

Author

Alois P. Heinz, Dec 16 2013

Keywords

• nonn

Examples

			a(2) = 4:
  .___.  .___.  .___.  .___.
  |_| |  | |_|  | ._|  |_. |
  |___|  |___|  |_|_|  |_|_| .
		

Links

• Wikipedia, Tromino

Crossrefs

Cf. A000105, A219874, A219952, A219975, A219994, A220061.

Programs

Maple

b:= proc(n, w, l) option remember; local k, t;
      if max(l[])>n then 0 elif n=0 then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, w, map(h->h-t, l))
    else for k while l[k]>0 do od;
         `if`(w, b(n, false, s(k=1, l)), 0)+
         `if`(k>1 and l[k-1]=1, b(n, w, s(k=2, k-1=2, l)), 0)+
         `if`(k b(n, evalb(irem(n, 3)>0), [0$n]): s:= subsop:
seq(a(n), n=0..10);

Mathematica

$RecursionLimit = 1000; s = ReplacePart; b[n_, w_, l_] := b[n, w, l] = Module[{k, t}, Which[Max[l] > n, 0,n == 0, 1,Min[l] > 0, t = Min[l]; b[n-t, w, l-t], True, For[k = 1, l[[k]] > 0, k++ ]; If[w, b[n, False, s[l, k -> 1]], 0]+If[k > 1 && l[[k-1]] == 1, b[n, w, s[l, {k -> 2, k-1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 1, b[n, w, s[l, {k -> 2, k+1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 0, b[n, w, s[l, {k -> 1, k+1 -> 2}]]+b[n, w, s[l, {k -> 2, k+1 -> 1}]] + If[w, b[n, False, s[l, {k -> 2, k+1 -> 2}]], 0], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, w, s[l, {k -> 2, k+1 -> 2, k+2 -> 2}]], 0] ] ]; a[n_] := b[n, Mod[n, 3] > 0, Array[0 &, n]]; Table[Print[an = a[n]]; an, {n, 0, 16}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)

Extensions

a(17) from Alois P. Heinz, Sep 24 2014