p[0]=1 p[1]=1/3
p[n+1]=2/3-p[n]/3
p[n]=(1/2){(-1/3)^n+1}

a[n]={p[n]-(1/3)^n}/2+(1/3)^n
=(1/4){(-1/3)^n+1}+(1/2)(1/3)^n
=(1/4){1+(2+(-1)^n)/3^n}

a[1]=1/3
a[2]=1/3
a[3]=7/27
a[4]=7/27
a[5]=61/243
a[6]=61/243