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
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