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

Author

R. H. Hardin, Jan 25 2002

Keywords

• nonn
• nice
• hard

Comments

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

Examples

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

References

• 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.

Links

• 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.

Crossrefs

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

Main diagonal of A219741 and A226444.

Programs

Maple

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

Mathematica

$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 *)

Formula

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

Extensions

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