First fix j and k, then we get
S1=∞∑i=0i≠j,k13i+j+k where j≠k
So we get S1=(32×13j+k)−132j+k−13j+2k
So now
S2=∞∑j=0j≠k((32×13j+k)−132j+k−13j+2k)
So S2=(94×13k)−(98×13k)−(32×19k)−(32×132k)+(233k)
Finally
S=∞∑k=0(94×13k)−(98×13k)−(32×19k)+∞∑k=0233k−∞∑k=03232k
we get
S=2716−2716+5426−2716 hence
S=81208
S1=∞∑i=0i≠j,k13i+j+k where j≠k
So we get S1=(32×13j+k)−132j+k−13j+2k
So now
S2=∞∑j=0j≠k((32×13j+k)−132j+k−13j+2k)
So S2=(94×13k)−(98×13k)−(32×19k)−(32×132k)+(233k)
Finally
S=∞∑k=0(94×13k)−(98×13k)−(32×19k)+∞∑k=0233k−∞∑k=03232k
we get
S=2716−2716+5426−2716 hence
S=81208