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A243608 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. 6
1, 1, 1, 1, 1, 4, 1, 1, 9, 20, 11, 1, 1, 16, 87, 196, 176, 46, 2, 1, 25, 244, 1195, 3145, 4431, 3161, 1007, 111, 2, 1, 36, 545, 4544, 22969, 73098, 147502, 185744, 140288, 59140, 12313, 1046, 26, 1, 49, 1056, 13215, 106819, 587149, 2251309, 6082000, 11562155 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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

LINKS

Alois P. Heinz, Rows n = 0..21, flattened

EXAMPLE

T(3,1) = 4:

  ._____.   ._____.   ._____.   ._____.

  | |_|_|   |_|_|_|   |_| |_|   |_|_|_|

  |___|_|   | |_|_|   |_|___|   |_| |_|

  |_|_|_|   |___|_|   |_|_|_|   |_|___|

T(4,4) = 1:

  ._______.

  | |_| |_|

  |___|___|

  | |_| |_|

  |___|___|

T(5,6) = 2:

  ._________.   ._________.

  | |_|_| |_|   |_| |_| |_|

  |___| |___|   | |___|___|

  |_| |___|_|   |___|_| |_|

  | |___| |_|   | |_| |___|

  |___|_|___|   |___|___|_| .

Triangle T(n,k) begins:

  1;

  1;

  1,  1;

  1,  4,   1;

  1,  9,  20,   11,    1;

  1, 16,  87,  196,  176,   46,    2;

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

MAPLE

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

      if n<2 then 1

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

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

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

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

      fi

    end:

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

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

MATHEMATICA

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

Table[T[n], {n, 0, 10}] // Flatten (* Jean-Fran├žois Alcover, Apr 12 2017, translated from Maple *)

CROSSREFS

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

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

Sequence in context: A126065 A299427 A126062 * A219207 A157108 A056647

Adjacent sequences:  A243605 A243606 A243607 * A243609 A243610 A243611

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Jun 07 2014

STATUS

approved

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Last modified June 17 05:50 EDT 2021. Contains 345080 sequences. (Running on oeis4.)