A075463 a(n) is the number of rotation-reflection inequivalent solutions to the all-ones lights out problem on an n X n square.

1, 1, 1, 5, 1, 1, 1, 1, 43, 1, 10, 1, 1, 5, 1, 43, 1, 1, 8356, 1, 1, 1, 2080, 5, 1, 1, 1, 1, 136, 131720, 1, 131720, 8356, 5, 10, 1, 1, 1, 536911936, 1, 1, 1, 1, 5, 1, 1, 134225920, 1, 43, 43, 1, 1, 1, 5, 1, 1, 1, 1, 524800, 1, 137439609088, 2099728, 1, 33564704, 549756338176, 1, 536911936, 1, 43, 1, 2080, 1, 1, 5, 1, 1, 1, 1, 2305843011898064896

Offset: 1

Author

Eric W. Weisstein, Sep 17 2002

Keywords

• nonn
• nice
• hard

Comments

Reflected and rotated solutions are considered identical.

References

• See A075462 for references.

Links

• Zhao Hui Du, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Lights Out Puzzle

Crossrefs

Cf. A075462, A075464.

Programs

Mathematica

a[n_, i_, j_ ] := Table[If[Total[Abs[{i, j} - {r, s}]] <= 1, 1, 0], {r, n}, {s, n}] //Flatten
a[n_, k_ ] := a[n, Quotient[k + n - 1, n], Mod[k, n, 1]]
m[n_ ] := a[n, # ] & /@ Range[n^2]
ker[n_ ] := NullSpace[m[n], Modulus -> 2]
b[n_ ] := Table[1, {n^2}]
sol[n_ ] := LinearSolve[m[n], b[n], Modulus -> 2];
allSolutions[n_ ] := Module[{s, k},
s = sol[n];
k = ker[n];
Mod[(s + # ) & /@ (Total[(#*k)] & /@ Tuples[{0, 1}, Length[k]]), 2]
] //Sort
MatrixRotate[m_ ] := Transpose[Reverse[m]]
MatrixRotate[m_, n_ ] := Nest[MatrixRotate, m, Mod[n, 4]]
DihedralOrbit[m_ ] := Union@Join[
MatrixRotate[m, # ] & /@ Range[0, 3],
MatrixRotate[Reverse[m], # ] & /@ Range[0, 3]
]
essentialSolutions[n_ ] := Module[{as},
as = Partition[ #, n] & /@ allSolutions[n];
Union[as, SameTest -> (MemberQ[DihedralOrbit[ #1], #2] &)]
]
Length[essentialSolutions[ # ]] & /@ Range[16]
(* Jacob A. Siehler *)

PARI

H(n)={
   my(r);
   r=matrix(n,n,X,Y,Mod(0,2));
   for(u=1,n,r[u,u]=Mod(1,2));
   for(u=1,n-1,r[u,u+1]=Mod(1,2);r[u+1,u]=Mod(1,2));
   r
}
E(n)={
   my(r);
   r=matrix(n,n,X,Y,Mod(0,2));
   for(u=1,n,r[u,u]=Mod(1,2));
   r
}
b(n)={
   vector(n,X,Mod(1,2))~
}
a(n)={
   my(x0,x1,tmp,y,z,mH,vb,v0,v1,yb,zb);
   my(A159257,A075462,A075463,a2,a3,a4,a5,tm,tb);
   x0=E(n);mH=H(n);x1=mH;vb=b(n);
   y=matrix(n,n);yb=vector(n);
   z=matrix(n,n);zb=vector(n);
   v0=vb;v1=mH*vb+v0;
   y[1,]=x0[1,];yb[1]=Mod(0,2);
   y[2,]=x1[1,];yb[2]=v0[1];
   z[1,]=x0[n,];zb[1]=Mod(0,2);
   z[2,]=x1[n,];zb[2]=v0[n];
   for(u=2,n-1,
      tmp=mH*x1+x0;x0=x1;x1=tmp;
      y[u+1,]=x1[1,];z[u+1,]=x1[n,];
      tmp=mH*v1+v0+vb;v0=v1;v1=tmp;
      yb[u+1]=v0[1];zb[u+1]=v0[n]
   );
   x1=mH*x1+x0;
   A159257 = n-matrank(x1);
   A075462=2^A159257;
   tm=matrix(2*n,n,X,Y,Mod(0,2));tb=vector(2*n,X,Mod(0,2))~;
   for(u=1,n,tm[u,]=x1[u,];tb[u]=v1[u]);
   for(u=1,n,
       tm[n+u,u]=Mod(1,2);tm[n+u,n-u+1]+=Mod(1,2);tb[n+u]=Mod(0,2)
   );
   a2=matinverseimage(tm,tb);
   if(length(a2)==0, a2=0, a2=2^(n-matrank(tm)));
   for(u=1,n,tm[n+u,]=y[u,];tb[n+u]=yb[u];tm[n+u,u]+=Mod(1,2));
   a3=matinverseimage(tm,tb);
   if(length(a3)==0, a3=0, a3=2^(n-matrank(tm)));
   for(u=1,n,tm[n+u,]=y[n+1-u,];tb[n+u]=yb[n+1-u];tm[n+u,u]+=Mod(1,2));
   a4=matinverseimage(tm,tb);
   if(length(a4)==0, a4=0, a4=2^(n-matrank(tm)));
   for(u=1,n,tm[n+u,]=y[u,]+z[n+1-u,];tb[n+u]=yb[u]+zb[n+1-u]);
   a5=matinverseimage(tm,tb);
   if(length(a5)==0, a5=0, a5=2^(n-matrank(tm)));
   A075463=(A075462+2*(a2+a3+a4)+a5)/8
} \\ Zhao Hui Du, Mar 29 2014

Extensions

a(19)-a(29) from Jacob A. Siehler, Apr 29 2008

Terms a(30) and beyond from Zhao Hui Du, Mar 24 2014