A322494 -id:A322494 - cp's OEIS frontend

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.

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, 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, 387, 524, 387, 129, 34, 1, 1, 1, 1, 55, 277, 1180, 2229, 2229, 1180, 277, 55, 1, 1

Offset: 0

Alois P. Heinz, Jun 06 2013

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

			A(3,3) = 6:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |_|_|_|  | |_|_|  |_|_|_|  |_| |_|  |_|_|_|  |_| |_|
  |_|_|_|  |___|_|  | |_|_|  |_|___|  |_| |_|  | |___|
  |_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|.
Square array A(n,k) begins:
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 1,  2,   3,    5,     8,     13,      21,       34, ...
  1, 1,  3,   6,   13,    28,     60,     129,      277, ...
  1, 1,  5,  13,   42,   126,    387,    1180,     3606, ...
  1, 1,  8,  28,  126,   524,   2229,    9425,    39905, ...
  1, 1, 13,  60,  387,  2229,  13322,   78661,   466288, ...
  1, 1, 21, 129, 1180,  9425,  78661,  647252,  5350080, ...
  1, 1, 34, 277, 3606, 39905, 466288, 5350080, 61758332, ...

Liang Kai, Antidiagonals n = 0..75, flattened (antidiagonals 0..44 from Alois P. Heinz)
Kai Liang, Independent Set Enumeration and Estimation of Related Constants of Grid Graphs and Their Variants, arXiv:2507.04007 [math.CO], 2025. See p. 13.

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.
Main diagonal gives A066864(n-1).
See A219741 for an array with very similar entries. - N. J. A. Sloane, Aug 22 2013
Cf. A322494.

b:= proc(n, l) option remember; local k, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; b(n, subsop(k=1, l))+
        `if`(k b(max(n, k), [0$min(n, k)]):
seq(seq(A(n, d-n), n=0..d), d=0..14);
[Zeilberger gives Maple code to find generating functions for the columns - see links in A228285. - N. J. A. Sloane, Aug 22 2013]

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 *)

The k-th column satisfies a recurrence of order Fibonacci(k+1) [Zeilberger] - see links in A228285. - N. J. A. Sloane, Aug 22 2013

A322496 Number of tilings of a 3 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

Original entry on oeis.org

1, 1, 3, 8, 18, 44, 107, 257, 621, 1500, 3620, 8740, 21101, 50941, 122983, 296908, 716798, 1730504, 4177807, 10086117, 24350041, 58786200, 141922440, 342631080, 827184601, 1997000281, 4821185163, 11639370608, 28099926378, 67839223364, 163778373107

Offset: 0

Alois P. Heinz, Dec 12 2018

The shapes of the tiles are:
              ._.
       ._.    | |
  ..  | |.  | |..
  ||  |__|  |___|  .

			a(3) = 8:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |_|_|_|  | |_|_|  |_|_|_|  |_| |_|  |_|_|_|  |_| |_|  | |_|_|  | | |_|
  |_|_|_|  |___|_|  | |_|_|  |_|___|  |_| |_|  | |___|  | |_|_|  | |___|
  |_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|  |_____|  |_____|  .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Polyomino
Index entries for linear recurrences with constant coefficients, signature (1,2,3,1).

Column k=3 of A322494.

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|3|2|1>>^n)[4$2]:
seq(a(n), n=0..40);

G.f.: -1/((x^2+x+1)*(x^2+2*x-1)).

a(n) = 2*a(n-1) + a(n-2) + A049347(n). - Greg Dresden, May 18 2020

A322497 Number of tilings of a 4 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

Original entry on oeis.org

1, 1, 5, 18, 68, 233, 838, 2989, 10687, 38097, 136002, 485370, 1732377, 6182628, 22065919, 78752901, 281068809, 1003130814, 3580164896, 12777572157, 45603031014, 162756761629, 580877276331, 2073145244569, 7399034871398, 26407082201462, 94246615039341

Offset: 0

Alois P. Heinz, Dec 12 2018

The shapes of the tiles are:
                       ._.
              ._.      | |
       ._.    | |      | |
  ..  | |.  | |..  | |.._.
  ||  |__|  |___|  |_____|  .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Polyomino
Index entries for linear recurrences with constant coefficients, signature (1,5,9,17,13,5,-2,-2).

Column k=4 of A322494.

LinearRecurrence[{1,5,9,17,13,5,-2,-2},{1,1,5,18,68,233,838,2989},30] (* Harvey P. Dale, May 02 2022 *)

G.f.: -(x+1)*(x^3+x-1)/((2*x^2+1)*(x^6+x^5-3*x^4-7*x^3-7*x^2-x+1)).

A322498 Number of tilings of a 5 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

Original entry on oeis.org

1, 1, 8, 44, 233, 1262, 6523, 34468, 181615, 957006, 5044388, 26575335, 140039124, 737911089, 3888300180, 20488828781, 107962314409, 568889946804, 2997672175041, 15795742092745, 83233076938962, 438583048406589, 2311041500385152, 12177654397383350

Offset: 0

Alois P. Heinz, Dec 12 2018

The shapes of the tiles are:
                       ._.
              ._.      | |
       ._.    | |      | |
  ..  | |.  | |..  | |.._.
  ||  |__|  |___|  |_____|  ... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Polyomino
Index entries for linear recurrences with constant coefficients, signature (1,10,36,93,240,392,551,576,361,324,58,58,57,-76,50,-10).

Column k=5 of A322494.

G.f.: (2*x^11 -8*x^10 +4*x^9 -7*x^8 -18*x^7 -29*x^6 -32*x^5 -20*x^4 -10*x^3 -3*x^2+1) / (10*x^16 -50*x^15 +76*x^14 -57*x^13 -58*x^12 -58*x^11 -324*x^10 -361*x^9 -576*x^8 -551*x^7 -392*x^6 -240*x^5 -93*x^4 -36*x^3 -10*x^2 -x+1).

A322499 Number of tilings of a 6 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

Original entry on oeis.org

1, 1, 13, 107, 838, 6523, 51420, 396500, 3086898, 24009733, 186735416, 1452320952, 11295996925, 87852065113, 683271518863, 5314139902766, 41330688551925, 321449136734050, 2500069029481412, 19444268928224485, 151227677898412290, 1176172285095334802

Offset: 0

Alois P. Heinz, Dec 12 2018

The shapes of the tiles are:
                       ._.
              ._.      | |
       ._.    | |      | |
  ..  | |.  | |..  | |.._.
  ||  |__|  |___|  |_____|  ... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Polyomino

Column k=6 of A322494.

A322500 Number of tilings of a 7 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

Original entry on oeis.org

1, 1, 21, 257, 2989, 34468, 396500, 4577274, 52338705, 600923375, 6897569624, 79170705722, 908721438900, 10430285080567, 119719520002653, 1374133015203548, 15772280213569812, 181033948036338279, 2077904397828614231, 23850149155575310594, 273751580233289217307

Offset: 0

Alois P. Heinz, Dec 12 2018

nonn

The shapes of the tiles are:
                       ._.
              ._.      | |
       ._.    | |      | |
  ..  | |.  | |..  | |.._.
  ||  |__|  |___|  |_____|  ... .

Alois P. Heinz, Table of n, a(n) for n = 0..944
Wikipedia, Polyomino

Column k=7 of A322494.

A322501 Number of tilings of an 8 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

Original entry on oeis.org

1, 1, 34, 621, 10687, 181615, 3086898, 52338705, 888837716, 15029968375, 254605651987, 4312608660046, 73048456842772, 1237312902210617, 20957926357019583, 354990566329457044, 6012925190578975735, 101848368521145389471, 1725133116479186659346

Offset: 0

Alois P. Heinz, Dec 12 2018

nonn

The shapes of the tiles are:
                       ._.
              ._.      | |
       ._.    | |      | |
  ..  | |.  | |..  | |.._.
  ||  |__|  |___|  |_____|  ... .

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Polyomino

Column k=8 of A322494.

A322502 Number of tilings of a 9 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

Original entry on oeis.org

1, 1, 55, 1500, 38097, 957006, 24009733, 600923375, 15029968375, 376116199534, 9394097589045, 234816663728554, 5869407851857410, 146709570371173411, 3667093829512515427, 91661210468678400305, 2291126428942263984999, 57268064504720859650768

Offset: 0

Alois P. Heinz, Dec 12 2018

nonn

The shapes of the tiles are:
                       ._.
              ._.      | |
       ._.    | |      | |
  ..  | |.  | |..  | |.._.
  ||  |__|  |___|  |_____|  ... .

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Polyomino

Column k=9 of A322494.

A322503 Number of tilings of a 10 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

Original entry on oeis.org

1, 1, 89, 3620, 136002, 5044388, 186735416, 6897569624, 254605651987, 9394097589045, 346688563051200, 12783317143959681, 471522273648618022, 17392356171712222007, 641526227823336378247, 23663017416745429407437, 872822255075124794048661

Offset: 0

Alois P. Heinz, Dec 12 2018

nonn

The shapes of the tiles are:
                       ._.
              ._.      | |
       ._.    | |      | |
  ..  | |.  | |..  | |.._.
  ||  |__|  |___|  |_____|  ... .

Alois P. Heinz, Table of n, a(n) for n = 0..40
Wikipedia, Polyomino

Column k=10 of A322494.

A322495 Number of tilings of an n X n square using V (2m+1)-ominoes (m >= 0) in standard orientation.

Original entry on oeis.org

1, 1, 2, 8, 68, 1262, 51420, 4577274, 888837716, 376116199534, 346688563051200, 695975307003529228

Offset: 0

Alois P. Heinz, Dec 12 2018

The shapes of the tiles are:
                       ._.
              ._.      | |
       ._.    | |      | |
  ..  | |.  | |..  | |.._.
  ||  |__|  |___|  |_____|  ... .

			a(3) = 8:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |_|_|_|  | |_|_|  |_|_|_|  |_| |_|  |_|_|_|  |_| |_|  | |_|_|  | | |_|
  |_|_|_|  |___|_|  | |_|_|  |_|___|  |_| |_|  | |___|  | |_|_|  | |___|
  |_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|  |_____|  |_____|.
.

Wikipedia, Polyomino

Main diagonal of A322494.

b:= proc(n, l) option remember; local k, m, r;
      if n=0 or l=[] then 1
    elif min(l)>0 then (t-> b(n-t, map(h->h-t, l)))(min(l))
    elif l[-1]=n then b(n, subsop(-1=[][], l))
    else for k while l[k]>0 do od; r:= 0;
         for m from 0 while k+m<=nops(l) and l[k+m]=0 and n>m do
           r:= r+b(n, [l[1..k-1][], 1$m, m+1, l[k+m+1..nops(l)][]])
         od; r
      fi
    end:
a:= n-> b(n, [0$n]):
seq(a(n), n=0..9);

b[n_, l_] := b[n, l] = Module[{k, m, r}, Which[n == 0 || l == {}, 1, Min[l] > 0, Function[t, b[n-t, l-t]][Min[l]], l[[-1]] == n, b[n, ReplacePart[l, -1 -> Nothing]], True, For[k = 1, l[[k]] > 0, k++]; r = 0; For[m = 0, k + m <= Length[l] && l[[k+m]] == 0 && n > m, m++, r = r + b[n, Join[l[[1 ;; k-1]], Array[1&, m], {m+1}, l[[k+m+1 ;; Length[l]]]]]]; r]];
a[n_] := b[n, Array[0&, n]];
a /@ Range[0, 9] (* Jean-François Alcover, Apr 22 2021, after Alois P. Heinz *)

cp's OEIS Frontend

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.

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A322496 Number of tilings of a 3 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

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A322497 Number of tilings of a 4 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

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A322498 Number of tilings of a 5 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

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A322499 Number of tilings of a 6 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

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A322500 Number of tilings of a 7 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

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A322501 Number of tilings of an 8 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

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A322502 Number of tilings of a 9 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

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A322503 Number of tilings of a 10 X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation.

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A322495 Number of tilings of an n X n square using V (2m+1)-ominoes (m >= 0) in standard orientation.

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