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

%I A066864 #82 Sep 13 2023 08:28:51
%S A066864 1,2,6,42,524,13322,647252,61758332,11435477118,4129523869606,
%T A066864 2902264461628298,3973109800760143708,10590895512774862686570,
%U A066864 54979738656662942307796576,555797909644630436677137498230,10941698340065066230952215658836402,419471520990343359533179780148504998680
%N A066864 Number of binary arrangements without adjacent 1's on n X n rhombic hexagonal grid.
%C A066864 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:
%C A066864   ._____   _____   _____   _____   _____   _____
%C A066864   |_|_|_| | |_|_| |_|_|_| |_| |_| |_|_|_| |_| |_|
%C A066864   |_|_|_| |___|_| | |_|_| |_|___| |_| |_| | |___|
%C A066864   |_|_|_| |_|_|_| |___|_| |_|_|_| |_|___| |___|_| - _Alois P. Heinz_, Jun 06 2013
%D A066864 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342-349.
%D A066864 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.
%H A066864 Vaclav Kotesovec and Alois P. Heinz, <a href="/A066864/b066864.txt">Table of n, a(n) for n = 0..28</a>
%H A066864 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/square/square.html">Hard Square Entropy Constant</a> [Broken link]
%H A066864 Steven R. Finch, <a href="http://web.archive.org/web/20010605012506/http://www.mathsoft.com/asolve/constant/square/square.html">Hard Square Entropy Constant</a> [From the Wayback machine]
%H A066864 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, pp. 69-71.
%F A066864 Limit_{n->oo} a(n)^(1/n^2) = 1.395485972... (see A085851).
%e A066864 Neighbors for n=4:
%e A066864   o--o--o--o
%e A066864   | /| /| /|
%e A066864   |/ |/ |/ |
%e A066864   o--o--o--o
%e A066864   | /| /| /|
%e A066864   |/ |/ |/ |
%e A066864   o--o--o--o
%e A066864   | /| /| /|
%e A066864   |/ |/ |/ |
%e A066864   o--o--o--o
%p A066864 a:= proc(n) option remember; local b; b:=
%p A066864       proc(n, l) option remember; local k;
%p A066864         if n<2 then 1
%p A066864       elif min(l[])>0 then b(n-1, map(h->h-1, l))
%p A066864       else for k while l[k]>0 do od; b(n, subsop(k=1, l))+
%p A066864            `if`(n>1 and k<nops(l) and l[k+1]=0,
%p A066864                 b(n, subsop(k=2, k+1=1, l)), 0)
%p A066864         fi
%p A066864       end: forget(b);
%p A066864       b(n+1, [0$n+1])
%p A066864     end:
%p A066864 seq(a(n), n=0..15);  # _Alois P. Heinz_, Aug 26 2013
%t A066864 $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_ *)
%o A066864 [S/R] proc a
%o A066864 stvar $[N][N]:boolean
%o A066864 init $[][] := false
%o A066864 cyset true
%o A066864 asgn $[][]->{false,true}
%o A066864 kill +[i in 0.. N-1](
%o A066864 +[j in 0.. N-1](
%o A066864 $[i][j]`*(
%o A066864 ($[i][j+1]`?(j<=N-2)|false)
%o A066864 +($[i-1][j+1]`?((i>0)*(j<=N-2))|false)
%o A066864 +($[i-1][j]`?(i>0)|false) ))) end
%Y A066864 Cf. A006506, A027683, A066863, A066865, A066866.
%Y A066864 Main diagonal of A219741 and A226444.
%K A066864 nonn,nice,hard
%O A066864 0,2
%A A066864 _R. H. Hardin_, Jan 25 2002
%E A066864 a(12)-a(21) from _Vaclav Kotesovec_, May 01 2012
%E A066864 a(0) and a(22) from _Alois P. Heinz_, Aug 26 2013
%E A066864 a(23) from _Alois P. Heinz_, Aug 28 2013
%E A066864 a(24) from _Vaclav Kotesovec_, Sep 19 2014
%E A066864 a(25) from _Alois P. Heinz_, Dec 03 2014
%E A066864 a(26)-a(28) from _Vaclav Kotesovec_, Aug 13 2016