[seqfan] GCD equals 1 pattern almost satisfies this recurrence.

Mats Granvik mgranvik at abo.fi
Mon Jun 13 21:22:55 CEST 2011


I am posting the same recurrence as earlier. Using the GCD=1 pattern as input
in the upper triangular half of the square array a, one gets almost  
the same pattern in the lower triangular half. Whenever a=1 then  
GCD(row,column)=1.
The sequence phi - 1 = A000010 - 1 is found where the columns divides  
the rows in a.

Mats Granvik

Mathematica program:

Clear[t, n, k, a, b, c];
nn = 20;
t[n_, 1] = If[n >= 1, 1, 0];
t[n_, k_] :=
   t[n, k] =
    If[n < k, If[GCD[n, k] == 1, 1, 0],
     Sum[t[n - i, k - 1] - t[n - i, k], {i, 1, k - 1}], 0];
a = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
MatrixForm[a]
b = Table[Table[DirichletCharacter[i, 1, n], {n, 1, nn}], {i, 1, nn}];
c = b - a;
MatrixForm[c]
Table[i - Total[c[[i]]], {i, 1, nn}]





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