

A006506


Number of n X n binary matrices with no 2 adjacent 1's, or number of configurations of nonattacking princes on an n X n board, where a "prince" attacks the four adjacent (nondiagonal) squares. Also number of independent vertex sets in an n X n grid.
(Formerly M1816)


36



1, 2, 7, 63, 1234, 55447, 5598861, 1280128950, 660647962955, 770548397261707, 2030049051145980050, 12083401651433651945979, 162481813349792588536582997, 4935961285224791538367780371090, 338752110195939290445247645371206783, 52521741712869136440040654451875316861275
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OFFSET

0,2


COMMENTS

A twodimensional generalization of the Fibonacci numbers.
Also the number of vertex covers in the n X n grid graph P_n X P_n.
A181030 (Number of n X n binary matrices with no leading bitstring in any row or column divisible by 4) is the same sequence. Proof from Steve Butler, Jan 26 2015: This is trivially true. A181030 is equivalent to this sequence by interchanging the roles of 0 and 1. In particular, A181030 looks for binary matrices with no leading bitstring divisible by 4, but a bitstring is divisible by 4 if and only if its last two digits is 0; in a binary matrix this can only be avoided if there are no two adjacent 0's (i.e., for any two adjacent 0's take the bitstring starting in that row or column and we are done); the present sequence looks for no two adjacent 1's. Similar reasons show that the array A181031 is equivalent to the array A089980.
Let R(n) be the set of squares that have vertices at integer coordinates and lie in the region of the plane x+y <= n+1, and let S(n) be the set of squares that have vertices at integer coordinates and lie in the region of the plane x+y1/2 <= n+2. Further let T be the collection of rectangular tiles with dimensions i X 1 or 1 X i with i arbitrary. Then a(2n) is the number of ways to tile R(n) using tiles from T and a(2n+1) is the number of ways to tile S(n) using tiles from T. (Note R(n) is the Aztec diamond of order n.)  Steve Butler, Jan 26 2015


REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342349.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



FORMULA

Limit_{n>oo} a(n)^(1/n^2) = c1 = 1.50304... is the hard square entropy constant A085850.  Benoit Cloitre, Nov 16 2003
a(n) appears to behave like A * c3^n * c1^(n^2) where c1 is as above, c3 = 1.143519129587 approximately, A = 1.0660826 approximately. This is based on numerical analysis of a(n) for n up to 19.  Brendan McKay, Nov 16 2003


MAPLE

A006506 := proc(N) local i, j, p, q; p := 1+x11;
if n=0 then return 1 fi;
for i from 2 to N do
q := pselect(has, p, x(i1)1);
p := p+expand(q*xi1)
od;
for j from 2 to N do
q := pselect(has, p, x1(j1));
p := subs(x1(j1)=1, p)+expand(q*x1j);
for i from 2 to N do
q := pselect(has, p, {x(i1)j, xi(j1)});
p := subs(xi(j1)=1, p)+expand(q*xij);
od
od;
map(icontent, p)
end:


MATHEMATICA

a[n_] := a[n] = (p = 1 + x[1, 1]; Do[q = p  Select[p, ! FreeQ[#, x[i1, 1]] &]; p = p + Expand[q*x[i, 1]], {i, 2, n}]; Do[q = p  Select[p, ! FreeQ[#, x[1, j1]] &]; p = (p /. x[i, j1] :> 1) + Expand[q*x[1, j]]; Do[q = p  Select[ p, ! FreeQ[#, x[i1, j]]  ! FreeQ[#, x[i, j1]] &]; p = (p /. x[i, j1] :> 1) + Expand[q*x[i, j]], {i, 2, n}], {j, 2, n}]; p /. x[_, _] > 1); a /@ Range[14] (* JeanFrançois Alcover, May 25 2011, after Maple prog. *)
Table[With[{g = GridGraph[{n, n}]}, Count[Subsets[Range[n^2], Length @ First @ FindIndependentVertexSet[g]], _?(IndependentVertexSetQ[g, #] &)]], {n, 5}] (* Eric W. Weisstein, May 28 2017 *)


PROG

(PARI) a(n)=L=fibonacci(n+2); p=v=vector(L, i, 1); c=0; for(i=0, 2^n1, j=i; while(j&&j%4<3, j\=2); if(j%4<3, p[c++]=i)); for(i=2, n, w=vector(L, j, 0); for(j=1, L, for(k=1, L, if(bitand(p[j], p[k])==0, w[j]+=v[k]))); v=w); sum(i=1, L, v[i]) \\ Robert Gerbicz, Jun 17 2011


CROSSREFS

Table of values for n x m matrices: A089934.
Cf. A232833 for refinement by number of 1's.


KEYWORD

nonn,nice,hard


AUTHOR



EXTENSIONS

Maple program updated and sequence extended by Robert Israel, Jun 16 2011


STATUS

approved



