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%I #21 May 17 2022 13:21:17
%S 1,1,4,0,16,128,162,34528,943096,1193600,3525377600,480585761344,
%T 2033502499954,46983507796973152,32908187880881958736,
%U 458324092996867592192,83153202122213272708832688,299769486068040749617049301344
%N Number of tilings of an n X n square using right trominoes and at most one monomino.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tromino">Tromino</a>
%e a(2) = 4:
%e .___. .___. .___. .___.
%e |_| | | |_| | ._| |_. |
%e |___| |___| |_|_| |_|_| .
%p b:= proc(n, w, l) option remember; local k, t;
%p if max(l[])>n then 0 elif n=0 then 1
%p elif min(l[])>0 then t:=min(l[]); b(n-t, w, map(h->h-t, l))
%p else for k while l[k]>0 do od;
%p `if`(w, b(n, false, s(k=1, l)), 0)+
%p `if`(k>1 and l[k-1]=1, b(n, w, s(k=2, k-1=2, l)), 0)+
%p `if`(k<nops(l) and l[k+1]=1, b(n, w, s(k=2, k+1=2, l)), 0)+
%p `if`(k<nops(l) and l[k+1]=0, b(n, w, s(k=1, k+1=2, l))+
%p b(n, w, s(k=2, k+1=1, l))+
%p `if`(w, b(n, false, s(k=2, k+1=2, l)), 0), 0)+
%p `if`(k+1<nops(l) and l[k+1]=0 and l[k+2]=0,
%p b(n, w, s(k=2, k+1=2, k+2=2, l)), 0)
%p fi
%p end:
%p a:= n-> b(n, evalb(irem(n, 3)>0), [0$n]): s:= subsop:
%p seq(a(n), n=0..10);
%t $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 *)
%Y Cf. A000105, A219874, A219952, A219975, A219994, A220061.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Dec 16 2013
%E a(17) from _Alois P. Heinz_, Sep 24 2014
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