[seqfan] Re: Mobius function modulo 3

[Charles Greathouse]

I see.  So more like

t[n_, 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];
M[n_]:=Inverse[Table[Table[t[s, u], {u, 1, n}], {s, 1, n}]][[All,1]];
MM[n_]:=Mod[M[n]+1,3]-1;
Test[n_]:=MM[n]==Table[MoebiusMu[k],{k,1,n}];

It is very laborious to check these values; I did verify the first 100 terms.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Fri, Jun 10, 2011 at 12:00 PM, Mats Granvik <mgranvik at abo.fi> wrote:
> For nn = 13, if I also add the line: t[13, 1] = 1; then I get 427472 =
> 3*142490.66666666666667. 427472 mod 3 equals 2 --> mu(13)=-1.
>
> Mats
>
>
> 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 = 13;
> 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[13, 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]
>
>
>
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