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A226444 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and L-tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A226444 #42 Sep 03 2025 13:54:23
%S A226444 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,3,1,1,1,1,5,6,5,1,1,1,1,8,13,13,
%T A226444 8,1,1,1,1,13,28,42,28,13,1,1,1,1,21,60,126,126,60,21,1,1,1,1,34,129,
%U A226444 387,524,387,129,34,1,1,1,1,55,277,1180,2229,2229,1180,277,55,1,1
%N A226444 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and L-tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A226444 An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed.
%H A226444 Liang Kai, <a href="/A226444/b226444.txt">Antidiagonals n = 0..75, flattened</a> (antidiagonals 0..44 from Alois P. Heinz)
%H A226444 Kai Liang, <a href="https://arxiv.org/abs/2507.04007">Independent Set Enumeration and Estimation of Related Constants of Grid Graphs and Their Variants</a>, arXiv:2507.04007 [math.CO], 2025. See p. 13.
%F A226444 The k-th column satisfies a recurrence of order Fibonacci(k+1) [Zeilberger] - see links in A228285. - _N. J. A. Sloane_, Aug 22 2013
%e A226444 A(3,3) = 6:
%e A226444   ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
%e A226444   |_|_|_|  | |_|_|  |_|_|_|  |_| |_|  |_|_|_|  |_| |_|
%e A226444   |_|_|_|  |___|_|  | |_|_|  |_|___|  |_| |_|  | |___|
%e A226444   |_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|.
%e A226444 Square array A(n,k) begins:
%e A226444   1, 1,  1,   1,    1,     1,      1,       1,        1, ...
%e A226444   1, 1,  1,   1,    1,     1,      1,       1,        1, ...
%e A226444   1, 1,  2,   3,    5,     8,     13,      21,       34, ...
%e A226444   1, 1,  3,   6,   13,    28,     60,     129,      277, ...
%e A226444   1, 1,  5,  13,   42,   126,    387,    1180,     3606, ...
%e A226444   1, 1,  8,  28,  126,   524,   2229,    9425,    39905, ...
%e A226444   1, 1, 13,  60,  387,  2229,  13322,   78661,   466288, ...
%e A226444   1, 1, 21, 129, 1180,  9425,  78661,  647252,  5350080, ...
%e A226444   1, 1, 34, 277, 3606, 39905, 466288, 5350080, 61758332, ...
%p A226444 b:= proc(n, l) option remember; local k, t;
%p A226444       if max(l[])>n then 0 elif n=0 or l=[] then 1
%p A226444     elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
%p A226444     else for k do if l[k]=0 then break fi od; b(n, subsop(k=1, l))+
%p A226444         `if`(k<nops(l) and l[k+1]=0, b(n, subsop(k=1, k+1=2, l)), 0)
%p A226444       fi
%p A226444     end:
%p A226444 A:= (n, k)-> b(max(n, k), [0$min(n, k)]):
%p A226444 seq(seq(A(n, d-n), n=0..d), d=0..14);
%p A226444 [Zeilberger gives Maple code to find generating functions for the columns - see links in A228285. - _N. J. A. Sloane_, Aug 22 2013]
%t A226444 b[n_, l_] := b[n, l] = Module[{k, t}, Which[Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 2}]], 0] ] ]; a[n_, k_] := b[Max[n, k], Array[0&, Min[n, k]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 18 2013, translated from Maple *)
%Y A226444 Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A002478, A105262, A219737(n-1) for n>2, A219738 (n-1) for n>2, A219739(n-1) for n>1, A219740(n-1) for n>2, A226543, A226544.
%Y A226444 Main diagonal gives A066864(n-1).
%Y A226444 See A219741 for an array with very similar entries. - _N. J. A. Sloane_, Aug 22 2013
%Y A226444 Cf. A322494.
%K A226444 nonn,tabl,changed
%O A226444 0,13
%A A226444 _Alois P. Heinz_, Jun 06 2013