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%I #18 Dec 31 2013 12:38:24
%S 1,7,201,9787,379688,16512483,726964790,31549810845,1378740599284,
%T 60239603421159,2630166605483293,114886450998314920,
%U 5017916294582867990,219163121582772423673,9572435654283943792842,418094220600909382190818,18261053013117932038592765
%N Number of tilings of a 2 X 4 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.
%H Alois P. Heinz, <a href="/A233507/b233507.txt">Table of n, a(n) for n = 0..200</a>
%e a(1) = A219866(4,2) = A129682(4) = A219866(2,4) = A219862(2) = 7:
%e ._______. ._______. ._______. ._______.
%e |_____| | | |_____| | | | | | |___| | |
%e |_____|_| |_|_____| |_|_|_|_| |___|_|_|
%e ._______. ._______. ._______.
%e | |___| | | | |___| |___|___|
%e |_|___|_| |_|_|___| |___|___|.
%p b:= proc(n, l) option remember; local k, t; t:= min(l[]);
%p if n=0 then 1
%p elif t>0 then b(n-t, map(h->h-t, l))
%p else for k while l[k]>0 do od;
%p add(`if`(n>=j, b(n, s(k=j, l)), 0), j=2..3)+
%p `if`(k<=6 and l[k+2]=0, b(n, s(k=1, k+2=1, l)), 0)+
%p `if`(k<=4 and l[k+2]=0 and l[k+2*2]=0, b(n, s(k=1,
%p k+2=1, k+2*2=1, l)), 0)+ `if`(irem(k, 2)>0 and
%p l[k+1]=0, b(n, s(k=1, k+1=1, l)), 0)
%p fi
%p end:
%p a:=n-> b(n, [0$8]): s:= subsop:
%p seq(a(n), n=0..10);
%t b[n_, l_] := b[n, l] = Module[{k, t}, t = Min[l]; Which[n == 0, 1, t > 0, b[n-t, l-t], True, For[k = 1, l[[k]] > 0, k++]; Sum[If[n >= j, b[n, ReplacePart[l, k -> j]], 0], {j, 2, 3}] + If[k <= 6 && l[[k + 2]] == 0, b[n, ReplacePart[l, {k -> 1, k+2 -> 1}]], 0] + If[k <= 4 && l[[k+2]] == 0 && l[[k+2*2]] == 0, b[n, ReplacePart[l, {k -> 1, k+2 -> 1, k+2*2 -> 1}]], 0] + If[Mod[k, 2] > 0 && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]], 0]]]; a[n_] := b[n, Array[0&, 8]]; Table[Print[an = a[n]]; an, {n, 0, 16}] (* _Jean-François Alcover_, Dec 30 2013, translated from Maple *)
%Y Cf. A000931, A129682, A219866, A219867, A233313, A233505, A233506, A233509.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Dec 11 2013
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