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A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A251072 #18 Jan 15 2019 19:15:53
%S A251072 1,1,1,1,1,1,1,1,1,1,1,1,13,1,1,1,1,1,41,1,1,1,1,1,19,281,1,1,1,1,1,1,
%T A251072 57,1183,1,1,1,1,1,1,26,121,6728,1,1,1,1,1,1,1,75,783,31529,1,1,1,1,1,
%U A251072 1,1,34,154,2861,167089,1,1,1,1,1,1,1,1,95,269,8133,817991,1,1
%N A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A251072 A(n,n) = A034856(n+2) for n>=2.
%H A251072 Alois P. Heinz, <a href="/A251072/b251072.txt">Antidiagonals n = 0..35, flattened</a>
%H A251072 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyomino">Polyomino</a>
%e A251072 Square array A(n,k) begins:
%e A251072   1, 1,      1,    1,    1,   1,   1,   1,  1, ...
%e A251072   1, 1,      1,    1,    1,   1,   1,   1,  1, ...
%e A251072   1, 1,     13,    1,    1,   1,   1,   1,  1, ...
%e A251072   1, 1,     41,   19,    1,   1,   1,   1,  1, ...
%e A251072   1, 1,    281,   57,   26,   1,   1,   1,  1, ...
%e A251072   1, 1,   1183,  121,   75,  34,   1,   1,  1, ...
%e A251072   1, 1,   6728,  783,  154,  95,  43,   1,  1, ...
%e A251072   1, 1,  31529, 2861,  269, 190, 117,  53,  1, ...
%e A251072   1, 1, 167089, 8133, 1732, 325, 229, 141, 64, ...
%p A251072 b:= proc(n, l) option remember; local d, k; d:= nops(l)/3;
%p A251072       if n=0 then 1
%p A251072     elif min(l[])>0 then (m->b(n-m, map(x->x-m, l)))(min(l[]))
%p A251072     else for k while l[k]>0 do od;
%p A251072          `if`(n<d, 0, b(n, subsop(k=d, l)))+
%p A251072          `if`(d=1 or k>2*d+1 or max(l[k..k+d-1][])>0, 0,
%p A251072           b(n, [l[1..k-1][], 1$d, l[k+d..3*d][]]))
%p A251072       fi
%p A251072     end:
%p A251072 A:= (n, k)-> `if`(k=0, 1, b(n, [0$3*k])):
%p A251072 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A251072 b[n_, l_List] := b[n, l] = Module[{d = Length[l]/3, k}, Which[n == 0, 1,  Min[l] > 0, Function[{m}, b[n-m, l-m]][Min[l]], True, For[k=1, l[[k]] > 0 , k++]; If[n<d, 0, b[n, ReplacePart[l, k -> d]]] + If[d == 1 || k > 2d + 1 || Max[l[[k ;; k + d - 1]]] > 0,  0,  b[n, Join[l[[1 ;; k-1]], Array[1&, d],  l[[k+d ;; 3*d]]]]]]]; A[n_, k_] := If[k == 0, 1, b[n, Array[0&, 3k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 30 2015, after _Alois P. Heinz_ *)
%Y A251072 Columns k=0+1,2-10 give: A000012, A028468, A251073, A251074, A247218, A251075, A251076, A251077, A251078, A251079.
%Y A251072 Cf. A034856, A250662.
%K A251072 nonn,tabl
%O A251072 0,13
%A A251072 _Alois P. Heinz_, Nov 29 2014