[seqfan] Mobius function modulo 3.

[Mats Granvik]

This Mathematica program produces a sequence 1,-1,2,-6,20,-68... which  
modulo 3 is equal to the mobius function 1,-1,-1,0,-1,1... modulo 3:  
1,2,2,0,2,1...

Clear[t, n, k, a, b, c, d, e, f, g, h, i, j, k, l, s, u, mat1, mat2,
   aa, bb, nn, signmatrix];
nn = 12;
t[1, 1] = 1;
t[2, 1] = 1;
t[3, 1] = 1;
t[4, 1] = 1;
t[5, 1] = 1;
t[6, 1] = 1;
t[7, 1] = 1;
t[8, 1] = 1;
t[9, 1] = 1;
t[10, 1] = 1;
t[11, 1] = 1;
t[12, 1] = 1;
t[s_, u_] :=
   t[s, u] =
    If[And[s > 1, u > 1],
     Sum[t[s - q, u - 1] + 2*t[s - q, u], {q, 1, u - 1}], 0];
mat1 = Table[Table[t[s, u], {u, 1, nn}], {s, 1, nn}];
mat2 = Inverse[mat1];
MatrixForm[mat2]
aa = Mod[mat2, 3];
MatrixForm[aa]
mat1[[1]][[nn]] = mat1[[nn]][[nn]];
mat1[[nn]][[nn]] = 0;
MatrixForm[mat1];
bb = Det[mat1];
Mod[bb, 3]
MatrixForm[(-((-2)^aa) + 1)/3]

Mats Granvik