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A276171 Irregular triangle T(n,k) read by rows: The number of tilings of the n X n board by 1 X 1 and k 3 X 3 squares, n >= 0, k >= 0.

%I A276171 #33 Oct 13 2023 10:02:01
%S A276171 1,1,1,1,1,1,4,1,9,1,16,30,12,1,1,25,132,200,79,1,36,360,1232,1246,1,
%T A276171 49,780,5048,13211,11984,4526,758,51,1,1,64,1470,15468,78851,193672,
%U A276171 234394,139188,37760,3600,1,81,2520,38972,324721,1490562,3761236,5052890,3305328,807648
%N A276171 Irregular triangle T(n,k) read by rows: The number of tilings of the n X n board by 1 X 1 and k 3 X 3 squares, n >= 0, k >= 0.
%C A276171 The number of monomers (1 X 1 squares) is n^2 - 9*k.
%C A276171 Row lengths are triplicated A002522.
%H A276171 Alois P. Heinz, <a href="/A276171/b276171.txt">Rows n = 0..24, flattened</a>
%H A276171 R. J. Mathar, <a href="http://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).
%F A276171 T(n,0) = 1.
%F A276171 T(n,1) = (n-2)^2, n >= 2.
%F A276171 From _Alois P. Heinz_, Sep 27 2016: (Start)
%F A276171 T(n,2) = (n-3)*(n-4)*(n-5)*(n+4)/2 for n > 2.
%F A276171 T(n,3) = (n^6 - 12*n^5 - 15*n^4 + 620*n^3 - 1234*n^2 - 7616*n + 22368)/6 for n > 5. (End)
%e A276171 The triangle starts in row n=0 with columns k=0,1,...:
%e A276171   1;
%e A276171   1;
%e A276171   1;
%e A276171   1,  1;
%e A276171   1,  4;
%e A276171   1,  9;
%e A276171   1, 16,  30,   12,     1;
%e A276171   1, 25, 132,  200,    79;
%e A276171   1, 36, 360, 1232,  1246;
%e A276171   1, 49, 780, 5048, 13211, 11984, 4526, 758, 51, 1;
%p A276171 b:= proc(n, l) option remember; local k, m; m:=min(l[]);
%p A276171       if n<3 then 1
%p A276171     elif m>0 then b(n-m, map(x->x-m, l))
%p A276171     else for k while l[k]>0 do od; b(n, subsop(k=1, l))+
%p A276171          expand(`if`(k+1<nops(l) and l[k+1..k+2]=[0$2],
%p A276171               b(n, subsop(k=3, k+1=3, k+2=3, l))*x, 0))
%p A276171       fi
%p A276171     end:
%p A276171 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):
%p A276171 seq(T(n), n=0..12);  # _Alois P. Heinz_, Sep 27 2016
%t A276171 b[n_, l_] := b[n, l] = Module[{ k, m}, m = Min[l]; Which[n < 3, 1, m > 0, b[n - m, l - m],  True, k = 1; While[l[[k]] > 0, k++]; b[n, ReplacePart[ l, k -> 1]] + Expand[If[k + 1 < Length[l] && l[[k+1 ;; k+2]] == {0, 0}, b[n, ReplacePart[l, {k -> 3, k+1 -> 3, k+2 -> 3}]]*x, 0]]]];
%t A276171 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]} ] ][b[n, Table[0, n]]];
%t A276171 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Nov 10 2017, after _Alois P. Heinz_ *)
%Y A276171 Cf. A002522, A140304 (row sums), A193580 (1 X 1 and 2 X 2 squares).
%K A276171 nonn,tabf
%O A276171 0,7
%A A276171 _R. J. Mathar_, Aug 23 2016