A066865 -id:A066865 - 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

A066866 Number of binary arrangements without adjacent 1's in n X n rhombic hexagonal grid torus.

Original entry on oeis.org

1, 5, 22, 201, 4216, 162314, 12329633, 1831137521, 528106112383, 296848246952000, 324932515409958655, 692572885398506075946, 2874785146216927021053015, 23237716875160177498526082523, 365789982527236500400197753931927, 11212996751916827855636781928754023265

Offset: 1

R. H. Hardin, Jan 25 2002

			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.

Sean A. Irvine, Table of n, a(n) for n = 1..18
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)
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 73.

Cf. A006506, A027683, A066863, A066864, A066865.

Terms a(11)-a(12) from Vaclav Kotesovec, May 07 2012
a(13) from Vaclav Kotesovec, Aug 15 2016
a(14) from Vaclav Kotesovec, May 24 2021
a(15)-a(16) 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

A067015 Triangle T(n,k) (n>=1, rows have irregular lengths) giving number of arrangements of k nonattacking princes on an n X n staggered hexagonal torus board.

Original entry on oeis.org

1, 1, 4, 1, 9, 9, 3, 1, 16, 72, 96, 32, 1, 25, 225, 910, 1695, 1355, 440, 70, 5, 1, 36, 522, 3936, 16659, 39996, 52860, 36108, 11871, 1960, 270, 36, 3, 1, 49, 1029, 12166, 89551, 429058, 1361619, 2864113, 3946208, 3481835, 1904833, 621026, 116060, 12285, 693, 7

Offset: 1

R. H. Hardin, Feb 17 2002

A partition function for the staggered grid torus.

			1; 1,4; 1,9,9,3; 1,16,72,96,32; 1,25,225,910,1695,1355,440,70,5; ...

Sean A. Irvine, Table of n, a(n) for n = 1..413 (rows 1..15 flattened)

Row sums give A066865.

Offset corrected by Sean A. Irvine, Nov 29 2023

A067967 Number of binary arrangements without adjacent 1's on n X n hexagonal staggered torus shifted for odd n.

Original entry on oeis.org

1, 5, 19, 217, 4076, 164258, 12285477, 1834600977, 527717587843, 296979228487760, 324881629286870822, 692625866382651263578, 2874716493700380888930840, 23237986479606982160703729647, 365788614113216462103977935612524, 11213018647250714014261414954480048385

Offset: 1

R. H. Hardin, Feb 02 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.

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

For bent instead of shifted for odd n see A066865.

More terms from Sean A. Irvine, Nov 18 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A066866 Number of binary arrangements without adjacent 1's in n X n rhombic hexagonal grid torus.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A067015 Triangle T(n,k) (n>=1, rows have irregular lengths) giving number of arrangements of k nonattacking princes on an n X n staggered hexagonal torus board.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A067967 Number of binary arrangements without adjacent 1's on n X n hexagonal staggered torus shifted for odd n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions