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A245013 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and 2 X 2 squares; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A245013 #49 May 29 2025 11:10:16
%S A245013 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,3,1,1,1,1,5,5,5,1,1,1,1,8,11,11,
%T A245013 8,1,1,1,1,13,21,35,21,13,1,1,1,1,21,43,93,93,43,21,1,1,1,1,34,85,269,
%U A245013 314,269,85,34,1,1,1,1,55,171,747,1213,1213,747,171,55,1,1
%N A245013 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and 2 X 2 squares; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A245013 Liang Kai, <a href="/A245013/b245013.txt">Antidiagonals n = 0..80, flattened</a> (antidiagonals n = 0..45 from Alois P. Heinz)
%H A245013 Kai Liang, <a href="https://arxiv.org/abs/2505.12776">Independent Set Enumeration in King Graphs by Tensor Network Contractions</a>, arXiv:2505.12776 [math.CO], 2025. See p. 4.
%H A245013 R. J. Mathar, <a href="https://arxiv.org/abs/1609.03964">Tiling n x m rectangles with 1 x 1 and s x s squares</a>, arXiv:1609.03964 [math.CO], 2016.
%H A245013 Johan Nilsson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Nilsson/nilsson15.html">On Counting the Number of Tilings of a Rectangle with Squares of Size 1 and 2</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.2.
%e A245013 A(3,3) = 5:
%e A245013   ._._._.  .___._.  ._.___.  ._._._.  ._._._.
%e A245013   |_|_|_|  |   |_|  |_|   |  |_|_|_|  |_|_|_|
%e A245013   |_|_|_|  |___|_|  |_|___|  |_|   |  |   |_|
%e A245013   |_|_|_|  |_|_|_|  |_|_|_|  |_|___|  |___|_| .
%e A245013 Square array A(n,k) begins:
%e A245013   1, 1,  1,  1,   1,    1,     1,      1, ...
%e A245013   1, 1,  1,  1,   1,    1,     1,      1, ...
%e A245013   1, 1,  2,  3,   5,    8,    13,     21, ...
%e A245013   1, 1,  3,  5,  11,   21,    43,     85, ...
%e A245013   1, 1,  5, 11,  35,   93,   269,    747, ...
%e A245013   1, 1,  8, 21,  93,  314,  1213,   4375, ...
%e A245013   1, 1, 13, 43, 269, 1213,  6427,  31387, ...
%e A245013   1, 1, 21, 85, 747, 4375, 31387, 202841, ...
%p A245013 b:= proc(n, l) option remember; local m, k; m:= min(l[]);
%p A245013       if m>0 then b(n-m, map(x->x-m, l))
%p A245013     elif n=0 then 1
%p A245013     else for k while l[k]>0 do od; b(n, subsop(k=1, l))+
%p A245013          `if`(n>1 and k<nops(l) and l[k+1]=0,
%p A245013          b(n, subsop(k=2, k+1=2, l)), 0)
%p A245013       fi
%p A245013     end:
%p A245013 A:= (n, k)-> `if`(min(n, k)<2, 1, b(max(n, k), [0$min(n, k)])):
%p A245013 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A245013 b[n_, l_] := b[n, l] = Module[{m=Min[l], k}, If[m>0, b[n-m, l-m], If[n == 0, 1, k=Position[l, 0, 1, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[n>1 && k<Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0]]]]; A[n_, k_] := If[Min[n, k]<2, 1, b[Max[n, k], Table[0, {Min[n, k]}]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 11 2014, after _Alois P. Heinz_ *)
%Y A245013 Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A001045(n+1), A054854, A054855, A063650, A063651, A063652, A063653, A063654.
%Y A245013 Main diagonal gives A063443.
%K A245013 nonn,tabl
%O A245013 0,13
%A A245013 _Alois P. Heinz_, Sep 16 2014