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

0,3

Links

Table of n, a(n) for n=0..17.

Wikipedia, Tromino

Example

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

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<nops(l) and l[k+1]=1, b(n, w, s(k=2, k+1=2, l)), 0)+
         `if`(k<nops(l) and l[k+1]=0, b(n, w, s(k=1, k+1=2, l))+
                                      b(n, w, s(k=2, k+1=1, l))+
         `if`(w, b(n, false, s(k=2, k+1=2, l)), 0), 0)+
         `if`(k+1<nops(l) and l[k+1]=0 and l[k+2]=0,
              b(n, w, s(k=2, k+1=2, k+2=2, l)), 0)
      fi
    end:
a:= n-> 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 *)

Crossrefs

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

Sequence in context: A002979 A137279 A343473 * A302771 A167350 A215669

Adjacent sequences: A233804 A233805 A233806 * A233808 A233809 A233810

Keyword

nonn

Author

Alois P. Heinz, Dec 16 2013

Extensions

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

Status

approved