Generalization of Luschny's Alternating Harmonic numbers

Table[Log[2] + (1/2)*
   Cos[Pi*n]*(PolyGamma[n/2 + 1/2] - PolyGamma[n/2 + 1]), {n, 1, 12}]
N[%]
N[Accumulate[
  Flatten[Table[{1/(2*n - 1), -1/(2*n - 0)}, {n, 1, 12/2}]]]]
Chop[%% - %]

nn = 12;
Table[N[Log[3] + (1/3)*
    Cos[Pi*n*2]*(PolyGamma[n/3 + (Mod[2*n, 3] + 1)/3] +
      PolyGamma[n/3 + (Mod[2*n + 1, 3] + 1)/3] -
      2*PolyGamma[n/3 + (Mod[2*n + 2, 3] + 1)/3])], {n, 1, nn}]
N[Accumulate[
  Flatten[Table[{1/(3*n - 2), 1/(3*n - 1), -2/(3*n + 0)}, {n, 1,
     nn/3}]]]]
Chop[%% - %]

nn = 12;
Table[N[Log[4] +
   1/4*Cos[Pi*n*2]*(PolyGamma[n/4 + (Mod[3*n, 4] + 1)/4] +
      PolyGamma[n/4 + (Mod[3*n + 1, 4] + 1)/4] +
      PolyGamma[n/4 + (Mod[3*n + 2, 4] + 1)/4] -
      3 PolyGamma[n/4 + (Mod[3*n + 3, 4] + 1)/4])], {n, 1, nn}]
N[Accumulate[
  Flatten[Table[{1/(4*n - 3), 1/(4*n - 2),
     1/(4*n - 1), -3/(4*n + 0)}, {n, 1, nn/4}]]]]
Chop[%% - %]