A066864 -id:A066864 - 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

A219741 T(n,k) = Unmatched value maps: number of nXk binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nXk array.

Original entry on oeis.org

1, 2, 2, 4, 6, 4, 7, 13, 13, 7, 12, 28, 42, 28, 12, 21, 60, 126, 126, 60, 21, 37, 129, 387, 524, 387, 129, 37, 65, 277, 1180, 2229, 2229, 1180, 277, 65, 114, 595, 3606, 9425, 13322, 9425, 3606, 595, 114, 200, 1278, 11012, 39905, 78661, 78661, 39905, 11012, 1278, 200

Offset: 1

R. H. Hardin, Nov 26 2012

Table starts
...1.....2......4........7.........12...........21.............37
...2.....6.....13.......28.........60..........129............277
...4....13.....42......126........387.........1180...........3606
...7....28....126......524.......2229.........9425..........39905
..12....60....387.....2229......13322........78661.........466288
..21...129...1180.....9425......78661.......647252........5350080
..37...277...3606....39905.....466288......5350080.......61758332
..65...595..11012...168925....2760690.....44159095......711479843
.114..1278..33636...715072...16350693....364647622.....8201909757
.200..2745.102733..3027049...96830726...3010723330....94531063074
.351..5896.313781.12813931..573456240..24858935864..1089590912023
.616.12664.958384.54243509.3396136349.205253857220.12558669019786

			Some solutions for n=3 k=4
..0..0..0..0....1..0..0..1....0..0..1..0....0..0..1..0....0..0..0..1
..0..1..0..0....0..0..0..0....1..0..0..0....0..0..0..0....0..1..0..0
..0..0..0..0....0..1..0..1....0..0..0..1....1..0..0..1....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..337

Column 1 is A005251(n+2).
Column 2 is A002478(n+1).
Column 3 is A105262(n+1) for n>1.
Main diagonal is A066864.
See A226444 for an array with very similar entries. - N. J. A. Sloane, Aug 22 2013

Zeilberger's Maple code (see links in A228285) would presumably give recurrences for the columns of this array. - N. J. A. Sloane, Aug 22 2013

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.

Original entry on oeis.org

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 *)

A066865 Number of binary arrangements without adjacent 1's on n X n staggered hexagonal torus bent for odd n.

Original entry on oeis.org

1, 5, 22, 217, 4726, 164258, 14840533, 1834600977, 669877863205, 296979228487760, 434542100979981567, 692625866382651263578, 4053364289624915167879497, 23237986479606982160703729647, 543749373021017146939376423644362, 11213018647250714014261414954480048385

Offset: 1

R. H. Hardin, Jan 25 2002

nonn

			Neighbors for n=4:
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
\|/ | \|/ |
-o--o--o--o-
 | /|\ | /|\
Neighbors for n=5:
\|/ | \|/ | \|/
 o--o--o--o--o
/| /|\ | /|\ |\
\|/ | \|/ | \|/
 o--o--o--o--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.

Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]

Cf. A006506, A027683, A066863, A066864, A066866, A067967 (shifted instead of bent).

Row sums of A067015.

More terms from Sean A. Irvine, Nov 18 2023

A066863 Number of binary arrangements without adjacent 1's on n X n staggered hexagonal grid.

Original entry on oeis.org

2, 6, 43, 557, 14432, 719469, 70372090, 13351521479, 4941545691252, 3559349503024593, 4993739972681894885, 13642580224488264353504, 72582736229683196932680697, 751993955499337790653321567382, 15172223086707160824288341875907978

Offset: 1

R. H. Hardin, Jan 25 2002

nonn

			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.

Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]
Sean A. Irvine, Java program (github)
Eric Weisstein's World of Mathematics, Hard Hexagon Entropy Constant

Cf. A006506, A027683, A066864, A066865, A066866.

More terms from Sean A. Irvine, Nov 15 2023

A085851 Decimal expansion of hard hexagon entropy constant.

Original entry on oeis.org

1, 3, 9, 5, 4, 8, 5, 9, 7, 2, 4, 7, 9, 3, 0, 2, 7, 3, 5, 2, 2, 9, 5, 0, 0, 6, 6, 3, 5, 6, 6, 8, 8, 8, 0, 6, 8, 9, 5, 4, 1, 0, 3, 7, 2, 8, 1, 4, 4, 6, 6, 1, 1, 9, 0, 8, 1, 7, 4, 7, 2, 1, 5, 6, 1, 3, 5, 7, 6, 0, 8, 8, 0, 3, 5, 8, 6, 9, 7, 7, 7, 4, 6, 8, 9, 8, 3, 7, 8, 7, 3, 0, 8, 5, 2, 7, 5, 4, 2, 7, 9, 0

Offset: 1

Eric W. Weisstein, Jul 05 2003

			1.395485972...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.12, p. 343.

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69.
Eric Weisstein's World of Mathematics, Hard Hexagon Entropy Constant.

Cf. A085850, A247413, A066864.

RealDigits[N[Root[-32751691810479015985152 + 97143135277377575190528*#1^4 - 73347491183630103871488*#1^6 - 71220809441400405884928*#1^8 + 107155448150443388043264*#1^10 - 72405670285649161617408*#1^12 + 2958015038376958230528*#1^14 + 7449488310131083100160*#1^16 + 797726698866658379776*#1^18 + 2505062311720673792* #1^20 + 2013290651222784*#1^22 + 25937424601*#1^24 & , 2], 200]][[1]] (* Vaclav Kotesovec, Apr 03 2014 *)

A182562 Number of ways to place k non-attacking semi-knights on an n x n chessboard, sum over all k>=0.

Original entry on oeis.org

2, 16, 288, 11664, 1458000, 506250000, 414720000000, 869730877440000, 5045702916833280000, 77297454895962562560000, 3017525202366485003182080000, 307389127582207654481154908160000, 83016370640108703579427655610531840000, 58770343311359208383258439665073059266560000

Offset: 1

Vaclav Kotesovec, May 05 2012

nonn

Semi-knight is a semi-leaper [1,2]. Moves of a semi-knight are allowed only in [2,1] and [-2,-1]. See also semi-bishops (A187235).

Vaclav Kotesovec, Table of n, a(n) for n = 1..60
V. Kotesovec, Non-attacking chess pieces

Cf. A067962, A067966, A063443, A006506, A067965, A066864, A067963, A067964, A182563

Table[If[EvenQ[n],Fibonacci[n/2+2]^(n+2)*Product[Fibonacci[j+2]^4,{j,1,n/2-1}],Fibonacci[(n+1)/2+2]^((n+1)/2)*Fibonacci[(n-1)/2+2]^((n-1)/2)*Product[Fibonacci[j+2]^4,{j,1,(n-1)/2}]],{n,1,20}]

a(n) = F(n/2+2)^(n+2)*prod(j=1,n/2-1,F(j+2)^4) if n is even, F((n+1)/2+2)^((n+1)/2)*F((n-1)/2+2)^((n-1)/2)*prod(j=1,(n-1)/2,F(j+2)^4) if n is odd, where F(n) = A000045(n) is the n-th Fibonacci number.

a(n) is asymptotic to C^4*((1+sqrt(5))/2)^((n+2)*(n+4))/5^(3/2*(n+2)), where C=1.226742010720353244... is Fibonacci Factorial Constant, see A062073.

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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A219741 T(n,k) = Unmatched value maps: number of nXk binary arrays indicating the locations of corresponding elements not equal to any horizontal, vertical or antidiagonal neighbor in a random 0..1 nXk array.

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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.

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A066865 Number of binary arrangements without adjacent 1's on n X n staggered hexagonal torus bent for odd n.

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A066863 Number of binary arrangements without adjacent 1's on n X n staggered hexagonal grid.

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A085851 Decimal expansion of hard hexagon entropy constant.

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Mathematica

A182562 Number of ways to place k non-attacking semi-knights on an n x n chessboard, sum over all k>=0.

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