A219741 -id:A219741 - cp's OEIS frontend

A066864 Number of binary arrangements without adjacent 1's on n X n rhombic hexagonal grid.

1, 2, 6, 42, 524, 13322, 647252, 61758332, 11435477118, 4129523869606, 2902264461628298, 3973109800760143708, 10590895512774862686570, 54979738656662942307796576, 555797909644630436677137498230, 10941698340065066230952215658836402, 419471520990343359533179780148504998680

Offset: 0

R. H. Hardin, Jan 25 2002

Also the number of tilings of an (n+1) X (n+1) square using 1 X 1 squares and L-tiles. An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed. a(2) = 6:
  .___   ___   ___   ___   ___   ___
  |||_| | ||| |||_| || || |||_| || ||
  |||_| |_|_| | ||| ||__| || || | |_|
  |||_| |||_| |_|_| |||_| ||__| |_|_| - Alois P. Heinz, Jun 06 2013

			Neighbors for n=4:
  o--o--o--o
  | /| /| /|
  |/ |/ |/ |
  o--o--o--o
  | /| /| /|
  |/ |/ |/ |
  o--o--o--o
  | /| /| /|
  |/ |/ |/ |
  o--o--o--o

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342-349.
J. Katzenelson and R. P. Kurshan, S/R: A Language for Specifying Protocols and Other Coordinating Processes, pp. 286-292 in Proc. IEEE Conf. Comput. Comm., 1986.

Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 0..28
Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 69-71.

Cf. A006506, A027683, A066863, A066865, A066866.

Main diagonal of A219741 and A226444.

a:= proc(n) option remember; local b; 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; b(n, subsop(k=1, l))+
           `if`(n>1 and kAlois P. Heinz, Aug 26 2013

$RecursionLimit = 1000; a[n0_] := a[n0] = Module[{b}, b[n_, l_List] := 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++]; b[n, ReplacePart[l, k -> 1]] + If[n>1 && k 2, k+1 -> 1}]], 0]]];  b[n0+1, Array[0&, n0+1]]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

Limit_{n->oo} a(n)^(1/n^2) = 1.395485972... (see A085851).

a(12)-a(21) from Vaclav Kotesovec, May 01 2012
a(0) and a(22) from Alois P. Heinz, Aug 26 2013
a(23) from Alois P. Heinz, Aug 28 2013
a(24) from Vaclav Kotesovec, Sep 19 2014
a(25) from Alois P. Heinz, Dec 03 2014
a(26)-a(28) from Vaclav Kotesovec, Aug 13 2016

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.

Original entry on oeis.org

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

A219737 Unmatched value maps: number of n X 4 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 n X 4 array.

Original entry on oeis.org

7, 28, 126, 524, 2229, 9425, 39905, 168925, 715072, 3027049, 12813931, 54243509, 229621433, 972024617, 4114736810, 17418344167, 73734658344, 312130693269, 1321299533915, 5593273893746, 23677229915913, 100229530526756

Offset: 1

R. H. Hardin, Nov 26 2012

nonn

Column 4 of A219741.

			Some solutions for n=3:
..0..1..0..1....0..0..1..0....0..0..0..1....1..0..1..0....1..0..0..0
..0..0..0..0....1..0..0..0....1..0..0..0....0..0..0..0....0..0..0..0
..1..0..0..0....0..1..0..1....0..1..0..0....0..1..0..1....1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..130

Cf. A219741.

Empirical: a(n) = a(n-1) + 10*a(n-2) + 15*a(n-3) + 4*a(n-4) - 6*a(n-5) - a(n-6) + 3*a(n-7) - a(n-8) for n>9.
Zeilberger's Maple code (see links in A228285) would give a proof that this recurrence is correct. - N. J. A. Sloane, Aug 22 2013
G.f.: x*(1 + x)*(7 + 14*x + 14*x^2 - x^3 - 2*x^4 - 2*x^5 + 3*x^6 - x^7) / (1 - x - 10*x^2 - 15*x^3 - 4*x^4 + 6*x^5 + x^6 - 3*x^7 + x^8). - Colin Barker, Mar 12 2018

A219738 Unmatched value maps: number of nX5 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX5 array.

Original entry on oeis.org

12, 60, 387, 2229, 13322, 78661, 466288, 2760690, 16350693, 96830726, 573456240, 3396136349, 20112704280, 119112043349, 705408898268, 4177593432263, 24740667779362, 146519915909536, 867724589734469, 5138864287202152

Offset: 1

R. H. Hardin Nov 26 2012

nonn

Column 5 of A219741

			Some solutions for n=3
..0..1..0..0..0....0..1..0..0..0....1..0..1..0..0....0..0..1..0..0
..0..0..1..0..0....0..0..0..0..1....0..0..0..0..0....0..0..0..0..1
..1..0..0..1..0....0..1..0..0..0....1..0..1..0..0....0..1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..69

Empirical: a(n) = a(n-1) +21*a(n-2) +48*a(n-3) +14*a(n-4) -69*a(n-5) -38*a(n-6) +68*a(n-7) +13*a(n-8) -57*a(n-9) +37*a(n-10) -8*a(n-11) -2*a(n-12) +a(n-13) for n>14

Zeilberger's Maple code (see links in A228285) would give a proof that this recurrence is correct. - N. J. A. Sloane, Aug 22 2013

A219739 Unmatched value maps: number of nX6 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX6 array.

Original entry on oeis.org

21, 129, 1180, 9425, 78661, 647252, 5350080, 44159095, 364647622, 3010723330, 24858935864, 205253857220, 1694729679245, 13992960979394, 115536377026413, 953955009802438, 7876568121973188, 65034854910941849, 536976546612691621

Offset: 1

R. H. Hardin Nov 26 2012

nonn

Column 6 of A219741

			Some solutions for n=3
..1..0..1..0..1..0....0..0..1..0..0..0....0..0..0..0..0..0....0..0..1..0..0..1
..0..0..0..0..0..0....1..0..0..1..0..1....0..1..0..0..0..1....0..0..0..0..0..0
..1..0..0..1..0..0....0..0..0..0..0..0....0..0..0..1..0..0....0..1..0..1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..38

Zeilberger's Maple code (see links in A228285) would give a recurrence for this sequence. - N. J. A. Sloane, Aug 22 2013

A219740 Unmatched value maps: number of nX7 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX7 array.

Original entry on oeis.org

37, 277, 3606, 39905, 466288, 5350080, 61758332, 711479843, 8201909757, 94531063074, 1089590912023, 12558669019786, 144752526242487, 1668430514943073, 19230483410164823, 221652312986931867

Offset: 1

R. H. Hardin Nov 26 2012

nonn

Column 7 of A219741

			Some solutions for n=3
..0..0..1..0..0..0..1....1..0..0..0..0..0..1....0..0..1..0..0..1..0
..0..0..0..0..0..0..0....0..1..0..0..1..0..0....0..0..0..0..0..0..0
..1..0..1..0..0..0..1....0..0..0..0..0..1..0....0..1..0..1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..43

Zeilberger's Maple code (see links in A228285) would give a recurrence for this sequence. - N. J. A. Sloane, Aug 22 2013

cp's OEIS Frontend

A066864 Number of binary arrangements without adjacent 1's on n X n rhombic hexagonal grid.

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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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A219737 Unmatched value maps: number of n X 4 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 n X 4 array.

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A219738 Unmatched value maps: number of nX5 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX5 array.

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A219739 Unmatched value maps: number of nX6 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX6 array.

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A219740 Unmatched value maps: number of nX7 binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nX7 array.

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