A243608 - OEIS

%I #29 Jan 15 2019 19:21:01

%S 1,1,1,1,1,4,1,1,9,20,11,1,1,16,87,196,176,46,2,1,25,244,1195,3145,

%T 4431,3161,1007,111,2,1,36,545,4544,22969,73098,147502,185744,140288,

%U 59140,12313,1046,26,1,49,1056,13215,106819,587149,2251309,6082000,11562155

%N Number T(n,k) of ways k L-tiles can be placed on an n X n square; triangle T(n,k), n>=0, 0<=k<=A229093(n), read by rows.

%C An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed.

%H Alois P. Heinz, <a href="/A243608/b243608.txt">Rows n = 0..21, flattened</a>

%e T(3,1) = 4:

%e   ._____.   ._____.   ._____.   ._____.

%e   | |_|_|   |_|_|_|   |_| |_|   |_|_|_|

%e   |___|_|   | |_|_|   |_|___|   |_| |_|

%e   |_|_|_|   |___|_|   |_|_|_|   |_|___|

%e T(4,4) = 1:

%e   ._______.

%e   | |_| |_|

%e   |___|___|

%e   | |_| |_|

%e   |___|___|

%e T(5,6) = 2:

%e   ._________.   ._________.

%e   | |_|_| |_|   |_| |_| |_|

%e   |___| |___|   | |___|___|

%e   |_| |___|_|   |___|_| |_|

%e   | |___| |_|   | |_| |___|

%e   |___|_|___|   |___|___|_| .

%e Triangle T(n,k) begins:

%e   1;

%e   1;

%e   1,  1;

%e   1,  4,   1;

%e   1,  9,  20,   11,    1;

%e   1, 16,  87,  196,  176,   46,    2;

%e   1, 25, 244, 1195, 3145, 4431, 3161, 1007, 111, 2;

%p b:= proc(n, l) option remember; local k;

%p       if n<2 then 1

%p     elif min(l[])>0 then b(n-1, map(h->h-1, l))

%p     else for k while l[k]>0 do od; expand(

%p          b(n, subsop(k=1, l))+ `if`(n>1 and k<nops(l)

%p          and l[k+1]=0, x*b(n, subsop(k=2, k+1=1, l)), 0))

%p       fi

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):

%p seq(T(n), n=0..10);

%t b[n_, l_] := b[n, l] = Module[{k}, Which[n<2, 1, Min[l]>0, b[n-1, l-1], True, For[k = 1, l[[k]] > 0, k++]; Expand[b[n, ReplacePart[l, k -> 1]] + If[n>1 && k<Length[l] && l[[k+1]]==0, x*b[n, ReplacePart[l, {k -> 2, k+1 -> 1}]], 0]]]];

%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Table[0, {n}]]];

%t Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Apr 12 2017, translated from Maple *)

%Y Columns k=0-6 give: A000012, A000290(n-1) for n>0, A243645, A243646, A243647, A243648, A243649.

%Y Row sums give main diagonal of A226444 or A066864(n-1) for n>0.

%K nonn,tabf

%O 0,6

%A _Alois P. Heinz_, Jun 07 2014