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

0,2

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

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.

Formula

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

Example

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

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 k<nops(l) and l[k+1]=0,
                b(n, subsop(k=2, k+1=1, l)), 0)
        fi
      end: forget(b);
      b(n+1, [0$n+1])
    end:
seq(a(n), n=0..15);  # Alois 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<Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {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 *)

Prog

[S/R] proc a
stvar $[N][N]:boolean
init $[][] := false
cyset true
asgn $[][]->{false, true}
kill +[i in 0.. N-1](
+[j in 0.. N-1](
$[i][j]`*(
($[i][j+1]`?(j<=N-2)|false)
+($[i-1][j+1]`?((i>0)*(j<=N-2))|false)
+($[i-1][j]`?(i>0)|false) ))) end

Crossrefs

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

Main diagonal of A219741 and A226444.

Sequence in context: A161632 A115974 A179330 * A181737 A116896 A061062

Adjacent sequences: A066861 A066862 A066863 * A066865 A066866 A066867

Keyword

nonn,nice,hard

Author

R. H. Hardin, Jan 25 2002

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

Status

approved