A243645 -id:A243645 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Jun 07 2014

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

			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;

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

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.

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 (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):
seq(T(n), n=0..10);

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 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-François Alcover, Apr 12 2017, translated from Maple *)

A348134 Number of ways two L-tiles (with rotation) can be placed on an n X n square.

Original entry on oeis.org

0, 0, 22, 336, 1422, 3952, 8790, 16992, 29806, 48672, 75222, 111280, 158862, 220176, 297622, 393792, 511470, 653632, 823446, 1024272, 1259662, 1533360, 1849302, 2211616, 2624622, 3092832, 3620950, 4213872, 4876686, 5614672, 6433302, 7338240, 8335342, 9430656

Offset: 1

Nicolas Bělohoubek, Oct 02 2021

All terms are even, because groups of ways, which are connected by 90 degrees rotation symmetry, are made up from 4 or 2 ways, so the number of ways will be some 4m+2n, and 4m+2n is even.

			For a(1) and a(2) there are fewer squares on the main square then squares of the 2 L-tiles, so a(1) = a(2) = 0.

Nicolas Bělohoubek, Visualization of 3rd term
Nicolas Bělohoubek, 90° rotation groups for 3rd term
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A242856, A243645.

LinearRecurrence[{5,-10,10,-5,1},{0,0,22,336,1422,3952},40] (* Harvey P. Dale, Mar 04 2023 *)

a(n) = 2*(n - 2)*(4*n^3 - 8*n^2 - 19*n + 32) for n > 1.
G.f.: 2*x^3*(11 + 113*x - 19*x^2 - 9*x^3)/(1 - x)^5. - Stefano Spezia, Oct 03 2021
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Aug 05 2025

cp's OEIS Frontend

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.

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