cp's OEIS Frontend

A219967 Number A(n,k) of tilings of a k X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A219967 #38 Mar 30 2025 10:55:55
%S A219967 1,1,1,1,0,1,1,0,0,1,1,1,1,1,1,1,0,1,1,0,1,1,0,1,2,1,0,1,1,1,2,3,3,2,
%T A219967 1,1,1,0,2,4,3,4,2,0,1,1,0,3,8,8,8,8,3,0,1,1,1,4,13,21,28,21,13,4,1,1,
%U A219967 1,0,5,19,31,65,65,31,19,5,0,1,1,0,7,35,70,170,267,170,70,35,7,0,1
%N A219967 Number A(n,k) of tilings of a k X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A219967 Liang Kai, <a href="/A219967/b219967.txt">Antidiagonals n = 0..35, flattened</a> (Antidiagonals 0..27 from Alois P. Heinz)
%H A219967 Kai Liang, <a href="https://arxiv.org/abs/2503.17698">Solving tiling enumeration problems by tensor network contractions</a>, arXiv:2503.17698 [math.CO], 2025. See p. 25, Table 4.
%H A219967 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tromino">Tromino</a>
%e A219967 A(4,4) = 3, because there are 3 tilings of a 4 X 4 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
%e A219967   ._._____.  ._____._.  ._._._._.
%e A219967   | |_____|  |_____| |  | . | . |
%e A219967   | | . | |  | | . | |  |___|___|
%e A219967   |_|___| |  | |___|_|  | . | . |
%e A219967   |_____|_|  |_|_____|  |___|___| .
%e A219967 Square array A(n,k) begins:
%e A219967   1,  1,  1,  1,  1,   1,    1,     1,     1, ...
%e A219967   1,  0,  0,  1,  0,   0,    1,     0,     0, ...
%e A219967   1,  0,  1,  1,  1,   2,    2,     3,     4, ...
%e A219967   1,  1,  1,  2,  3,   4,    8,    13,    19, ...
%e A219967   1,  0,  1,  3,  3,   8,   21,    31,    70, ...
%e A219967   1,  0,  2,  4,  8,  28,   65,   170,   456, ...
%e A219967   1,  1,  2,  8, 21,  65,  267,   804,  2530, ...
%e A219967   1,  0,  3, 13, 31, 170,  804,  2744, 12343, ...
%e A219967   1,  0,  4, 19, 70, 456, 2530, 12343, 66653, ...
%p A219967 b:= proc(n, l) option remember; local k, t;
%p A219967       if max(l[])>n then 0 elif n=0 or l=[] then 1
%p A219967     elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
%p A219967     else for k do if l[k]=0 then break fi od;
%p A219967          b(n, subsop(k=3, l))+
%p A219967          `if`(k<nops(l) and l[k+1]=0, b(n, subsop(k=2, k+1=2, l)), 0)+
%p A219967          `if`(k+1<nops(l) and l[k+1]=0 and l[k+2]=0,
%p A219967             b(n, subsop(k=1, k+1=1, k+2=1, l)), 0)
%p A219967       fi
%p A219967     end:
%p A219967 A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
%p A219967 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A219967 b[n_, l_] := b[n, l] = Module[{ k, t}, If [Max[l] > n, 0, If[n == 0 || l == {}, 1, If[ Min[l] > 0 ,t = Min[l]; b[n-t, l-t], k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 3]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1, k+2 -> 1}]], 0] ] ] ] ]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 16 2013, translated from Maple *)
%Y A219967 Columns (or rows) k=0-10 give: A000012, A079978, A000931(n+3), A219968, A202536, A219969, A219970, A219971, A219972, A219973, A219974.
%Y A219967 Main diagonal gives: A219975.
%K A219967 nonn,tabl
%O A219967 0,25
%A A219967 _Alois P. Heinz_, Dec 02 2012